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  <div class="headertitle">
<div class="title">GenericPacketMathFunctions.h</div>  </div>
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<div class="contents">
<div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno">    1</span>&#160;<span class="comment">// This file is part of Eigen, a lightweight C++ template library</span></div>
<div class="line"><a name="l00002"></a><span class="lineno">    2</span>&#160;<span class="comment">// for linear algebra.</span></div>
<div class="line"><a name="l00003"></a><span class="lineno">    3</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00004"></a><span class="lineno">    4</span>&#160;<span class="comment">// Copyright (C) 2007 Julien Pommier</span></div>
<div class="line"><a name="l00005"></a><span class="lineno">    5</span>&#160;<span class="comment">// Copyright (C) 2014 Pedro Gonnet (pedro.gonnet@gmail.com)</span></div>
<div class="line"><a name="l00006"></a><span class="lineno">    6</span>&#160;<span class="comment">// Copyright (C) 2009-2019 Gael Guennebaud &lt;gael.guennebaud@inria.fr&gt;</span></div>
<div class="line"><a name="l00007"></a><span class="lineno">    7</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00008"></a><span class="lineno">    8</span>&#160;<span class="comment">// This Source Code Form is subject to the terms of the Mozilla</span></div>
<div class="line"><a name="l00009"></a><span class="lineno">    9</span>&#160;<span class="comment">// Public License v. 2.0. If a copy of the MPL was not distributed</span></div>
<div class="line"><a name="l00010"></a><span class="lineno">   10</span>&#160;<span class="comment">// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.</span></div>
<div class="line"><a name="l00011"></a><span class="lineno">   11</span>&#160; </div>
<div class="line"><a name="l00012"></a><span class="lineno">   12</span>&#160;<span class="comment">/* The exp and log functions of this file initially come from</span></div>
<div class="line"><a name="l00013"></a><span class="lineno">   13</span>&#160;<span class="comment"> * Julien Pommier&#39;s sse math library: http://gruntthepeon.free.fr/ssemath/</span></div>
<div class="line"><a name="l00014"></a><span class="lineno">   14</span>&#160;<span class="comment"> */</span></div>
<div class="line"><a name="l00015"></a><span class="lineno">   15</span>&#160; </div>
<div class="line"><a name="l00016"></a><span class="lineno">   16</span>&#160;<span class="preprocessor">#ifndef EIGEN_ARCH_GENERIC_PACKET_MATH_FUNCTIONS_H</span></div>
<div class="line"><a name="l00017"></a><span class="lineno">   17</span>&#160;<span class="preprocessor">#define EIGEN_ARCH_GENERIC_PACKET_MATH_FUNCTIONS_H</span></div>
<div class="line"><a name="l00018"></a><span class="lineno">   18</span>&#160; </div>
<div class="line"><a name="l00019"></a><span class="lineno">   19</span>&#160;<span class="preprocessor">#include &quot;../../InternalHeaderCheck.h&quot;</span></div>
<div class="line"><a name="l00020"></a><span class="lineno">   20</span>&#160; </div>
<div class="line"><a name="l00021"></a><span class="lineno">   21</span>&#160;<span class="keyword">namespace </span><a class="code" href="namespaceEigen.html">Eigen</a> {</div>
<div class="line"><a name="l00022"></a><span class="lineno">   22</span>&#160;<span class="keyword">namespace </span>internal {</div>
<div class="line"><a name="l00023"></a><span class="lineno">   23</span>&#160; </div>
<div class="line"><a name="l00024"></a><span class="lineno">   24</span>&#160;<span class="comment">// Creates a Scalar integer type with same bit-width.</span></div>
<div class="line"><a name="l00025"></a><span class="lineno">   25</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> T&gt; <span class="keyword">struct </span>make_integer;</div>
<div class="line"><a name="l00026"></a><span class="lineno">   26</span>&#160;<span class="keyword">template</span>&lt;&gt; <span class="keyword">struct </span>make_integer&lt;float&gt;    { <span class="keyword">typedef</span> numext::int32_t type; };</div>
<div class="line"><a name="l00027"></a><span class="lineno">   27</span>&#160;<span class="keyword">template</span>&lt;&gt; <span class="keyword">struct </span>make_integer&lt;double&gt;   { <span class="keyword">typedef</span> numext::int64_t type; };</div>
<div class="line"><a name="l00028"></a><span class="lineno">   28</span>&#160;<span class="keyword">template</span>&lt;&gt; <span class="keyword">struct </span>make_integer&lt;half&gt;     { <span class="keyword">typedef</span> numext::int16_t type; };</div>
<div class="line"><a name="l00029"></a><span class="lineno">   29</span>&#160;<span class="keyword">template</span>&lt;&gt; <span class="keyword">struct </span>make_integer&lt;bfloat16&gt; { <span class="keyword">typedef</span> numext::int16_t type; };</div>
<div class="line"><a name="l00030"></a><span class="lineno">   30</span>&#160; </div>
<div class="line"><a name="l00031"></a><span class="lineno">   31</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt; EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC</div>
<div class="line"><a name="l00032"></a><span class="lineno">   32</span>&#160;Packet pfrexp_generic_get_biased_exponent(<span class="keyword">const</span> Packet&amp; a) {</div>
<div class="line"><a name="l00033"></a><span class="lineno">   33</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::type Scalar;</div>
<div class="line"><a name="l00034"></a><span class="lineno">   34</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::integer_packet PacketI;</div>
<div class="line"><a name="l00035"></a><span class="lineno">   35</span>&#160;  <span class="keyword">static</span> constexpr <span class="keywordtype">int</span> mantissa_bits = numext::numeric_limits&lt;Scalar&gt;::digits - 1;</div>
<div class="line"><a name="l00036"></a><span class="lineno">   36</span>&#160;  <span class="keywordflow">return</span> pcast&lt;PacketI, Packet&gt;(plogical_shift_right&lt;mantissa_bits&gt;(preinterpret&lt;PacketI&gt;(pabs(a))));</div>
<div class="line"><a name="l00037"></a><span class="lineno">   37</span>&#160;}</div>
<div class="line"><a name="l00038"></a><span class="lineno">   38</span>&#160; </div>
<div class="line"><a name="l00039"></a><span class="lineno">   39</span>&#160;<span class="comment">// Safely applies frexp, correctly handles denormals.</span></div>
<div class="line"><a name="l00040"></a><span class="lineno">   40</span>&#160;<span class="comment">// Assumes IEEE floating point format.</span></div>
<div class="line"><a name="l00041"></a><span class="lineno">   41</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt; EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC</div>
<div class="line"><a name="l00042"></a><span class="lineno">   42</span>&#160;Packet pfrexp_generic(<span class="keyword">const</span> Packet&amp; a, Packet&amp; exponent) {</div>
<div class="line"><a name="l00043"></a><span class="lineno">   43</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::type Scalar;</div>
<div class="line"><a name="l00044"></a><span class="lineno">   44</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> make_unsigned&lt;typename make_integer&lt;Scalar&gt;::type&gt;::type ScalarUI;</div>
<div class="line"><a name="l00045"></a><span class="lineno">   45</span>&#160;  <span class="keyword">static</span> constexpr <span class="keywordtype">int</span></div>
<div class="line"><a name="l00046"></a><span class="lineno">   46</span>&#160;    TotalBits = <span class="keyword">sizeof</span>(Scalar) * CHAR_BIT,</div>
<div class="line"><a name="l00047"></a><span class="lineno">   47</span>&#160;    MantissaBits = numext::numeric_limits&lt;Scalar&gt;::digits - 1,</div>
<div class="line"><a name="l00048"></a><span class="lineno">   48</span>&#160;    ExponentBits = TotalBits - MantissaBits - 1;</div>
<div class="line"><a name="l00049"></a><span class="lineno">   49</span>&#160; </div>
<div class="line"><a name="l00050"></a><span class="lineno">   50</span>&#160;  EIGEN_CONSTEXPR ScalarUI scalar_sign_mantissa_mask =</div>
<div class="line"><a name="l00051"></a><span class="lineno">   51</span>&#160;      ~(((ScalarUI(1) &lt;&lt; ExponentBits) - ScalarUI(1)) &lt;&lt; MantissaBits); <span class="comment">// ~0x7f800000</span></div>
<div class="line"><a name="l00052"></a><span class="lineno">   52</span>&#160;  <span class="keyword">const</span> Packet sign_mantissa_mask = pset1frombits&lt;Packet&gt;(<span class="keyword">static_cast&lt;</span>ScalarUI<span class="keyword">&gt;</span>(scalar_sign_mantissa_mask));</div>
<div class="line"><a name="l00053"></a><span class="lineno">   53</span>&#160;  <span class="keyword">const</span> Packet half = pset1&lt;Packet&gt;(Scalar(0.5));</div>
<div class="line"><a name="l00054"></a><span class="lineno">   54</span>&#160;  <span class="keyword">const</span> Packet zero = pzero(a);</div>
<div class="line"><a name="l00055"></a><span class="lineno">   55</span>&#160;  <span class="keyword">const</span> Packet normal_min = pset1&lt;Packet&gt;((numext::numeric_limits&lt;Scalar&gt;::min)()); <span class="comment">// Minimum normal value, 2^-126</span></div>
<div class="line"><a name="l00056"></a><span class="lineno">   56</span>&#160; </div>
<div class="line"><a name="l00057"></a><span class="lineno">   57</span>&#160;  <span class="comment">// To handle denormals, normalize by multiplying by 2^(int(MantissaBits)+1).</span></div>
<div class="line"><a name="l00058"></a><span class="lineno">   58</span>&#160;  <span class="keyword">const</span> Packet is_denormal = pcmp_lt(pabs(a), normal_min);</div>
<div class="line"><a name="l00059"></a><span class="lineno">   59</span>&#160;  EIGEN_CONSTEXPR ScalarUI scalar_normalization_offset = ScalarUI(MantissaBits + 1); <span class="comment">// 24</span></div>
<div class="line"><a name="l00060"></a><span class="lineno">   60</span>&#160;  <span class="comment">// The following cannot be constexpr because bfloat16(uint16_t) is not constexpr.</span></div>
<div class="line"><a name="l00061"></a><span class="lineno">   61</span>&#160;  <span class="keyword">const</span> Scalar scalar_normalization_factor = Scalar(ScalarUI(1) &lt;&lt; <span class="keywordtype">int</span>(scalar_normalization_offset)); <span class="comment">// 2^24</span></div>
<div class="line"><a name="l00062"></a><span class="lineno">   62</span>&#160;  <span class="keyword">const</span> Packet normalization_factor = pset1&lt;Packet&gt;(scalar_normalization_factor);</div>
<div class="line"><a name="l00063"></a><span class="lineno">   63</span>&#160;  <span class="keyword">const</span> Packet normalized_a = pselect(is_denormal, pmul(a, normalization_factor), a);</div>
<div class="line"><a name="l00064"></a><span class="lineno">   64</span>&#160; </div>
<div class="line"><a name="l00065"></a><span class="lineno">   65</span>&#160;  <span class="comment">// Determine exponent offset: -126 if normal, -126-24 if denormal</span></div>
<div class="line"><a name="l00066"></a><span class="lineno">   66</span>&#160;  <span class="keyword">const</span> Scalar scalar_exponent_offset = -Scalar((ScalarUI(1)&lt;&lt;(ExponentBits-1)) - ScalarUI(2)); <span class="comment">// -126</span></div>
<div class="line"><a name="l00067"></a><span class="lineno">   67</span>&#160;  Packet exponent_offset = pset1&lt;Packet&gt;(scalar_exponent_offset);</div>
<div class="line"><a name="l00068"></a><span class="lineno">   68</span>&#160;  <span class="keyword">const</span> Packet normalization_offset = pset1&lt;Packet&gt;(-Scalar(scalar_normalization_offset)); <span class="comment">// -24</span></div>
<div class="line"><a name="l00069"></a><span class="lineno">   69</span>&#160;  exponent_offset = pselect(is_denormal, padd(exponent_offset, normalization_offset), exponent_offset);</div>
<div class="line"><a name="l00070"></a><span class="lineno">   70</span>&#160; </div>
<div class="line"><a name="l00071"></a><span class="lineno">   71</span>&#160;  <span class="comment">// Determine exponent and mantissa from normalized_a.</span></div>
<div class="line"><a name="l00072"></a><span class="lineno">   72</span>&#160;  exponent = pfrexp_generic_get_biased_exponent(normalized_a);</div>
<div class="line"><a name="l00073"></a><span class="lineno">   73</span>&#160;  <span class="comment">// Zero, Inf and NaN return &#39;a&#39; unmodified, exponent is zero</span></div>
<div class="line"><a name="l00074"></a><span class="lineno">   74</span>&#160;  <span class="comment">// (technically the exponent is unspecified for inf/NaN, but GCC/Clang set it to zero)</span></div>
<div class="line"><a name="l00075"></a><span class="lineno">   75</span>&#160;  <span class="keyword">const</span> Scalar scalar_non_finite_exponent = Scalar((ScalarUI(1) &lt;&lt; ExponentBits) - ScalarUI(1));  <span class="comment">// 255</span></div>
<div class="line"><a name="l00076"></a><span class="lineno">   76</span>&#160;  <span class="keyword">const</span> Packet non_finite_exponent = pset1&lt;Packet&gt;(scalar_non_finite_exponent);</div>
<div class="line"><a name="l00077"></a><span class="lineno">   77</span>&#160;  <span class="keyword">const</span> Packet is_zero_or_not_finite = por(pcmp_eq(a, zero), pcmp_eq(exponent, non_finite_exponent));</div>
<div class="line"><a name="l00078"></a><span class="lineno">   78</span>&#160;  <span class="keyword">const</span> Packet m = pselect(is_zero_or_not_finite, a, por(pand(normalized_a, sign_mantissa_mask), half));</div>
<div class="line"><a name="l00079"></a><span class="lineno">   79</span>&#160;  exponent = pselect(is_zero_or_not_finite, zero, padd(exponent, exponent_offset));</div>
<div class="line"><a name="l00080"></a><span class="lineno">   80</span>&#160;  <span class="keywordflow">return</span> m;</div>
<div class="line"><a name="l00081"></a><span class="lineno">   81</span>&#160;}</div>
<div class="line"><a name="l00082"></a><span class="lineno">   82</span>&#160; </div>
<div class="line"><a name="l00083"></a><span class="lineno">   83</span>&#160;<span class="comment">// Safely applies ldexp, correctly handles overflows, underflows and denormals.</span></div>
<div class="line"><a name="l00084"></a><span class="lineno">   84</span>&#160;<span class="comment">// Assumes IEEE floating point format.</span></div>
<div class="line"><a name="l00085"></a><span class="lineno">   85</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt; EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC</div>
<div class="line"><a name="l00086"></a><span class="lineno">   86</span>&#160;Packet pldexp_generic(<span class="keyword">const</span> Packet&amp; a, <span class="keyword">const</span> Packet&amp; exponent) {</div>
<div class="line"><a name="l00087"></a><span class="lineno">   87</span>&#160;  <span class="comment">// We want to return a * 2^exponent, allowing for all possible integer</span></div>
<div class="line"><a name="l00088"></a><span class="lineno">   88</span>&#160;  <span class="comment">// exponents without overflowing or underflowing in intermediate</span></div>
<div class="line"><a name="l00089"></a><span class="lineno">   89</span>&#160;  <span class="comment">// computations.</span></div>
<div class="line"><a name="l00090"></a><span class="lineno">   90</span>&#160;  <span class="comment">//</span></div>
<div class="line"><a name="l00091"></a><span class="lineno">   91</span>&#160;  <span class="comment">// Since &#39;a&#39; and the output can be denormal, the maximum range of &#39;exponent&#39;</span></div>
<div class="line"><a name="l00092"></a><span class="lineno">   92</span>&#160;  <span class="comment">// to consider for a float is:</span></div>
<div class="line"><a name="l00093"></a><span class="lineno">   93</span>&#160;  <span class="comment">//   -255-23 -&gt; 255+23</span></div>
<div class="line"><a name="l00094"></a><span class="lineno">   94</span>&#160;  <span class="comment">// Below -278 any finite float &#39;a&#39; will become zero, and above +278 any</span></div>
<div class="line"><a name="l00095"></a><span class="lineno">   95</span>&#160;  <span class="comment">// finite float will become inf, including when &#39;a&#39; is the smallest possible </span></div>
<div class="line"><a name="l00096"></a><span class="lineno">   96</span>&#160;  <span class="comment">// denormal.</span></div>
<div class="line"><a name="l00097"></a><span class="lineno">   97</span>&#160;  <span class="comment">//</span></div>
<div class="line"><a name="l00098"></a><span class="lineno">   98</span>&#160;  <span class="comment">// Unfortunately, 2^(278) cannot be represented using either one or two</span></div>
<div class="line"><a name="l00099"></a><span class="lineno">   99</span>&#160;  <span class="comment">// finite normal floats, so we must split the scale factor into at least</span></div>
<div class="line"><a name="l00100"></a><span class="lineno">  100</span>&#160;  <span class="comment">// three parts. It turns out to be faster to split &#39;exponent&#39; into four</span></div>
<div class="line"><a name="l00101"></a><span class="lineno">  101</span>&#160;  <span class="comment">// factors, since [exponent&gt;&gt;2] is much faster to compute that [exponent/3].</span></div>
<div class="line"><a name="l00102"></a><span class="lineno">  102</span>&#160;  <span class="comment">//</span></div>
<div class="line"><a name="l00103"></a><span class="lineno">  103</span>&#160;  <span class="comment">// Set e = min(max(exponent, -278), 278);</span></div>
<div class="line"><a name="l00104"></a><span class="lineno">  104</span>&#160;  <span class="comment">//     b = floor(e/4);</span></div>
<div class="line"><a name="l00105"></a><span class="lineno">  105</span>&#160;  <span class="comment">//   out = ((((a * 2^(b)) * 2^(b)) * 2^(b)) * 2^(e-3*b))</span></div>
<div class="line"><a name="l00106"></a><span class="lineno">  106</span>&#160;  <span class="comment">//</span></div>
<div class="line"><a name="l00107"></a><span class="lineno">  107</span>&#160;  <span class="comment">// This will avoid any intermediate overflows and correctly handle 0, inf,</span></div>
<div class="line"><a name="l00108"></a><span class="lineno">  108</span>&#160;  <span class="comment">// NaN cases.</span></div>
<div class="line"><a name="l00109"></a><span class="lineno">  109</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::integer_packet PacketI;</div>
<div class="line"><a name="l00110"></a><span class="lineno">  110</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::type Scalar;</div>
<div class="line"><a name="l00111"></a><span class="lineno">  111</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> unpacket_traits&lt;PacketI&gt;::type ScalarI;</div>
<div class="line"><a name="l00112"></a><span class="lineno">  112</span>&#160;  <span class="keyword">static</span> constexpr <span class="keywordtype">int</span></div>
<div class="line"><a name="l00113"></a><span class="lineno">  113</span>&#160;    TotalBits = <span class="keyword">sizeof</span>(Scalar) * CHAR_BIT,</div>
<div class="line"><a name="l00114"></a><span class="lineno">  114</span>&#160;    MantissaBits = numext::numeric_limits&lt;Scalar&gt;::digits - 1,</div>
<div class="line"><a name="l00115"></a><span class="lineno">  115</span>&#160;    ExponentBits = TotalBits - MantissaBits - 1;</div>
<div class="line"><a name="l00116"></a><span class="lineno">  116</span>&#160; </div>
<div class="line"><a name="l00117"></a><span class="lineno">  117</span>&#160;  <span class="keyword">const</span> Packet max_exponent = pset1&lt;Packet&gt;(Scalar((ScalarI(1)&lt;&lt;ExponentBits) + ScalarI(MantissaBits - 1)));  <span class="comment">// 278</span></div>
<div class="line"><a name="l00118"></a><span class="lineno">  118</span>&#160;  <span class="keyword">const</span> PacketI bias = pset1&lt;PacketI&gt;((ScalarI(1)&lt;&lt;(ExponentBits-1)) - ScalarI(1));  <span class="comment">// 127</span></div>
<div class="line"><a name="l00119"></a><span class="lineno">  119</span>&#160;  <span class="keyword">const</span> PacketI e = pcast&lt;Packet, PacketI&gt;(pmin(pmax(exponent, pnegate(max_exponent)), max_exponent));</div>
<div class="line"><a name="l00120"></a><span class="lineno">  120</span>&#160;  PacketI b = parithmetic_shift_right&lt;2&gt;(e); <span class="comment">// floor(e/4);</span></div>
<div class="line"><a name="l00121"></a><span class="lineno">  121</span>&#160;  Packet c = preinterpret&lt;Packet&gt;(plogical_shift_left&lt;MantissaBits&gt;(padd(b, bias)));  <span class="comment">// 2^b</span></div>
<div class="line"><a name="l00122"></a><span class="lineno">  122</span>&#160;  Packet out = pmul(pmul(pmul(a, c), c), c);  <span class="comment">// a * 2^(3b)</span></div>
<div class="line"><a name="l00123"></a><span class="lineno">  123</span>&#160;  b = psub(psub(psub(e, b), b), b); <span class="comment">// e - 3b</span></div>
<div class="line"><a name="l00124"></a><span class="lineno">  124</span>&#160;  c = preinterpret&lt;Packet&gt;(plogical_shift_left&lt;MantissaBits&gt;(padd(b, bias)));  <span class="comment">// 2^(e-3*b)</span></div>
<div class="line"><a name="l00125"></a><span class="lineno">  125</span>&#160;  out = pmul(out, c);</div>
<div class="line"><a name="l00126"></a><span class="lineno">  126</span>&#160;  <span class="keywordflow">return</span> out;</div>
<div class="line"><a name="l00127"></a><span class="lineno">  127</span>&#160;}</div>
<div class="line"><a name="l00128"></a><span class="lineno">  128</span>&#160; </div>
<div class="line"><a name="l00129"></a><span class="lineno">  129</span>&#160;<span class="comment">// Explicitly multiplies</span></div>
<div class="line"><a name="l00130"></a><span class="lineno">  130</span>&#160;<span class="comment">//    a * (2^e)</span></div>
<div class="line"><a name="l00131"></a><span class="lineno">  131</span>&#160;<span class="comment">// clamping e to the range</span></div>
<div class="line"><a name="l00132"></a><span class="lineno">  132</span>&#160;<span class="comment">// [NumTraits&lt;Scalar&gt;::min_exponent()-2, NumTraits&lt;Scalar&gt;::max_exponent()]</span></div>
<div class="line"><a name="l00133"></a><span class="lineno">  133</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00134"></a><span class="lineno">  134</span>&#160;<span class="comment">// This is approx 7x faster than pldexp_impl, but will prematurely over/underflow</span></div>
<div class="line"><a name="l00135"></a><span class="lineno">  135</span>&#160;<span class="comment">// if 2^e doesn&#39;t fit into a normal floating-point Scalar.</span></div>
<div class="line"><a name="l00136"></a><span class="lineno">  136</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00137"></a><span class="lineno">  137</span>&#160;<span class="comment">// Assumes IEEE floating point format</span></div>
<div class="line"><a name="l00138"></a><span class="lineno">  138</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00139"></a><span class="lineno">  139</span>&#160;<span class="keyword">struct </span>pldexp_fast_impl {</div>
<div class="line"><a name="l00140"></a><span class="lineno">  140</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::integer_packet PacketI;</div>
<div class="line"><a name="l00141"></a><span class="lineno">  141</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::type Scalar;</div>
<div class="line"><a name="l00142"></a><span class="lineno">  142</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> unpacket_traits&lt;PacketI&gt;::type ScalarI;</div>
<div class="line"><a name="l00143"></a><span class="lineno">  143</span>&#160;  <span class="keyword">static</span> constexpr <span class="keywordtype">int</span></div>
<div class="line"><a name="l00144"></a><span class="lineno">  144</span>&#160;    TotalBits = <span class="keyword">sizeof</span>(Scalar) * CHAR_BIT,</div>
<div class="line"><a name="l00145"></a><span class="lineno">  145</span>&#160;    MantissaBits = numext::numeric_limits&lt;Scalar&gt;::digits - 1,</div>
<div class="line"><a name="l00146"></a><span class="lineno">  146</span>&#160;    ExponentBits = TotalBits - MantissaBits - 1;</div>
<div class="line"><a name="l00147"></a><span class="lineno">  147</span>&#160; </div>
<div class="line"><a name="l00148"></a><span class="lineno">  148</span>&#160;  <span class="keyword">static</span> EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC</div>
<div class="line"><a name="l00149"></a><span class="lineno">  149</span>&#160;  Packet run(<span class="keyword">const</span> Packet&amp; a, <span class="keyword">const</span> Packet&amp; exponent) {</div>
<div class="line"><a name="l00150"></a><span class="lineno">  150</span>&#160;    <span class="keyword">const</span> Packet bias = pset1&lt;Packet&gt;(Scalar((ScalarI(1)&lt;&lt;(ExponentBits-1)) - ScalarI(1)));  <span class="comment">// 127</span></div>
<div class="line"><a name="l00151"></a><span class="lineno">  151</span>&#160;    <span class="keyword">const</span> Packet limit = pset1&lt;Packet&gt;(Scalar((ScalarI(1)&lt;&lt;ExponentBits) - ScalarI(1)));     <span class="comment">// 255</span></div>
<div class="line"><a name="l00152"></a><span class="lineno">  152</span>&#160;    <span class="comment">// restrict biased exponent between 0 and 255 for float.</span></div>
<div class="line"><a name="l00153"></a><span class="lineno">  153</span>&#160;    <span class="keyword">const</span> PacketI e = pcast&lt;Packet, PacketI&gt;(pmin(pmax(padd(exponent, bias), pzero(limit)), limit)); <span class="comment">// exponent + 127</span></div>
<div class="line"><a name="l00154"></a><span class="lineno">  154</span>&#160;    <span class="comment">// return a * (2^e)</span></div>
<div class="line"><a name="l00155"></a><span class="lineno">  155</span>&#160;    <span class="keywordflow">return</span> pmul(a, preinterpret&lt;Packet&gt;(plogical_shift_left&lt;MantissaBits&gt;(e)));</div>
<div class="line"><a name="l00156"></a><span class="lineno">  156</span>&#160;  }</div>
<div class="line"><a name="l00157"></a><span class="lineno">  157</span>&#160;};</div>
<div class="line"><a name="l00158"></a><span class="lineno">  158</span>&#160; </div>
<div class="line"><a name="l00159"></a><span class="lineno">  159</span>&#160;<span class="comment">// Natural or base 2 logarithm.</span></div>
<div class="line"><a name="l00160"></a><span class="lineno">  160</span>&#160;<span class="comment">// Computes log(x) as log(2^e * m) = C*e + log(m), where the constant C =log(2)</span></div>
<div class="line"><a name="l00161"></a><span class="lineno">  161</span>&#160;<span class="comment">// and m is in the range [sqrt(1/2),sqrt(2)). In this range, the logarithm can</span></div>
<div class="line"><a name="l00162"></a><span class="lineno">  162</span>&#160;<span class="comment">// be easily approximated by a polynomial centered on m=1 for stability.</span></div>
<div class="line"><a name="l00163"></a><span class="lineno">  163</span>&#160;<span class="comment">// TODO(gonnet): Further reduce the interval allowing for lower-degree</span></div>
<div class="line"><a name="l00164"></a><span class="lineno">  164</span>&#160;<span class="comment">//               polynomial interpolants -&gt; ... -&gt; profit!</span></div>
<div class="line"><a name="l00165"></a><span class="lineno">  165</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> Packet, <span class="keywordtype">bool</span> base2&gt;</div>
<div class="line"><a name="l00166"></a><span class="lineno">  166</span>&#160;EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS</div>
<div class="line"><a name="l00167"></a><span class="lineno">  167</span>&#160;Packet plog_impl_float(<span class="keyword">const</span> Packet _x)</div>
<div class="line"><a name="l00168"></a><span class="lineno">  168</span>&#160;{</div>
<div class="line"><a name="l00169"></a><span class="lineno">  169</span>&#160;  <span class="keyword">const</span> Packet cst_1              = pset1&lt;Packet&gt;(1.0f);</div>
<div class="line"><a name="l00170"></a><span class="lineno">  170</span>&#160;  <span class="keyword">const</span> Packet cst_minus_inf      = pset1frombits&lt;Packet&gt;(<span class="keyword">static_cast&lt;</span>Eigen::numext::uint32_t<span class="keyword">&gt;</span>(0xff800000u));</div>
<div class="line"><a name="l00171"></a><span class="lineno">  171</span>&#160;  <span class="keyword">const</span> Packet cst_pos_inf        = pset1frombits&lt;Packet&gt;(<span class="keyword">static_cast&lt;</span>Eigen::numext::uint32_t<span class="keyword">&gt;</span>(0x7f800000u));</div>
<div class="line"><a name="l00172"></a><span class="lineno">  172</span>&#160; </div>
<div class="line"><a name="l00173"></a><span class="lineno">  173</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_SQRTHF = pset1&lt;Packet&gt;(0.707106781186547524f);</div>
<div class="line"><a name="l00174"></a><span class="lineno">  174</span>&#160;  Packet e, x;</div>
<div class="line"><a name="l00175"></a><span class="lineno">  175</span>&#160;  <span class="comment">// extract significant in the range [0.5,1) and exponent</span></div>
<div class="line"><a name="l00176"></a><span class="lineno">  176</span>&#160;  x = pfrexp(_x,e);</div>
<div class="line"><a name="l00177"></a><span class="lineno">  177</span>&#160; </div>
<div class="line"><a name="l00178"></a><span class="lineno">  178</span>&#160;  <span class="comment">// part2: Shift the inputs from the range [0.5,1) to [sqrt(1/2),sqrt(2))</span></div>
<div class="line"><a name="l00179"></a><span class="lineno">  179</span>&#160;  <span class="comment">// and shift by -1. The values are then centered around 0, which improves</span></div>
<div class="line"><a name="l00180"></a><span class="lineno">  180</span>&#160;  <span class="comment">// the stability of the polynomial evaluation.</span></div>
<div class="line"><a name="l00181"></a><span class="lineno">  181</span>&#160;  <span class="comment">//   if( x &lt; SQRTHF ) {</span></div>
<div class="line"><a name="l00182"></a><span class="lineno">  182</span>&#160;  <span class="comment">//     e -= 1;</span></div>
<div class="line"><a name="l00183"></a><span class="lineno">  183</span>&#160;  <span class="comment">//     x = x + x - 1.0;</span></div>
<div class="line"><a name="l00184"></a><span class="lineno">  184</span>&#160;  <span class="comment">//   } else { x = x - 1.0; }</span></div>
<div class="line"><a name="l00185"></a><span class="lineno">  185</span>&#160;  Packet mask = pcmp_lt(x, cst_cephes_SQRTHF);</div>
<div class="line"><a name="l00186"></a><span class="lineno">  186</span>&#160;  Packet tmp = pand(x, mask);</div>
<div class="line"><a name="l00187"></a><span class="lineno">  187</span>&#160;  x = psub(x, cst_1);</div>
<div class="line"><a name="l00188"></a><span class="lineno">  188</span>&#160;  e = psub(e, pand(cst_1, mask));</div>
<div class="line"><a name="l00189"></a><span class="lineno">  189</span>&#160;  x = padd(x, tmp);</div>
<div class="line"><a name="l00190"></a><span class="lineno">  190</span>&#160; </div>
<div class="line"><a name="l00191"></a><span class="lineno">  191</span>&#160;  <span class="comment">// Polynomial coefficients for rational (3,3) r(x) = p(x)/q(x)</span></div>
<div class="line"><a name="l00192"></a><span class="lineno">  192</span>&#160;  <span class="comment">// approximating log(1+x) on [sqrt(0.5)-1;sqrt(2)-1].</span></div>
<div class="line"><a name="l00193"></a><span class="lineno">  193</span>&#160;  <span class="keyword">const</span> Packet cst_p1 = pset1&lt;Packet&gt;(1.0000000190281136f);</div>
<div class="line"><a name="l00194"></a><span class="lineno">  194</span>&#160;  <span class="keyword">const</span> Packet cst_p2 = pset1&lt;Packet&gt;(1.0000000190281063f);</div>
<div class="line"><a name="l00195"></a><span class="lineno">  195</span>&#160;  <span class="keyword">const</span> Packet cst_p3 = pset1&lt;Packet&gt;(0.18256296349849254f);</div>
<div class="line"><a name="l00196"></a><span class="lineno">  196</span>&#160;  <span class="keyword">const</span> Packet cst_q1 = pset1&lt;Packet&gt;(1.4999999999999927f);</div>
<div class="line"><a name="l00197"></a><span class="lineno">  197</span>&#160;  <span class="keyword">const</span> Packet cst_q2 = pset1&lt;Packet&gt;(0.59923249590823520f);</div>
<div class="line"><a name="l00198"></a><span class="lineno">  198</span>&#160;  <span class="keyword">const</span> Packet cst_q3 = pset1&lt;Packet&gt;(0.049616247954120038f);</div>
<div class="line"><a name="l00199"></a><span class="lineno">  199</span>&#160; </div>
<div class="line"><a name="l00200"></a><span class="lineno">  200</span>&#160;  Packet p = pmadd(x, cst_p3, cst_p2);</div>
<div class="line"><a name="l00201"></a><span class="lineno">  201</span>&#160;  p = pmadd(x, p, cst_p1);</div>
<div class="line"><a name="l00202"></a><span class="lineno">  202</span>&#160;  p = pmul(x, p);</div>
<div class="line"><a name="l00203"></a><span class="lineno">  203</span>&#160;  Packet q = pmadd(x, cst_q3, cst_q2);</div>
<div class="line"><a name="l00204"></a><span class="lineno">  204</span>&#160;  q = pmadd(x, q, cst_q1);</div>
<div class="line"><a name="l00205"></a><span class="lineno">  205</span>&#160;  q = pmadd(x, q, cst_1);</div>
<div class="line"><a name="l00206"></a><span class="lineno">  206</span>&#160;  x = pdiv(p, q);</div>
<div class="line"><a name="l00207"></a><span class="lineno">  207</span>&#160; </div>
<div class="line"><a name="l00208"></a><span class="lineno">  208</span>&#160;  <span class="comment">// Add the logarithm of the exponent back to the result of the interpolation.</span></div>
<div class="line"><a name="l00209"></a><span class="lineno">  209</span>&#160;  <span class="keywordflow">if</span> (base2) {</div>
<div class="line"><a name="l00210"></a><span class="lineno">  210</span>&#160;    <span class="keyword">const</span> Packet cst_log2e = pset1&lt;Packet&gt;(<span class="keyword">static_cast&lt;</span><span class="keywordtype">float</span><span class="keyword">&gt;</span>(EIGEN_LOG2E));</div>
<div class="line"><a name="l00211"></a><span class="lineno">  211</span>&#160;    x = pmadd(x, cst_log2e, e);</div>
<div class="line"><a name="l00212"></a><span class="lineno">  212</span>&#160;  } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00213"></a><span class="lineno">  213</span>&#160;    <span class="keyword">const</span> Packet cst_ln2 = pset1&lt;Packet&gt;(<span class="keyword">static_cast&lt;</span><span class="keywordtype">float</span><span class="keyword">&gt;</span>(EIGEN_LN2));</div>
<div class="line"><a name="l00214"></a><span class="lineno">  214</span>&#160;    x = pmadd(e, cst_ln2, x);</div>
<div class="line"><a name="l00215"></a><span class="lineno">  215</span>&#160;  }</div>
<div class="line"><a name="l00216"></a><span class="lineno">  216</span>&#160; </div>
<div class="line"><a name="l00217"></a><span class="lineno">  217</span>&#160;  Packet invalid_mask = pcmp_lt_or_nan(_x, pzero(_x));</div>
<div class="line"><a name="l00218"></a><span class="lineno">  218</span>&#160;  Packet iszero_mask  = pcmp_eq(_x,pzero(_x));</div>
<div class="line"><a name="l00219"></a><span class="lineno">  219</span>&#160;  Packet pos_inf_mask = pcmp_eq(_x,cst_pos_inf);</div>
<div class="line"><a name="l00220"></a><span class="lineno">  220</span>&#160;  <span class="comment">// Filter out invalid inputs, i.e.:</span></div>
<div class="line"><a name="l00221"></a><span class="lineno">  221</span>&#160;  <span class="comment">//  - negative arg will be NAN</span></div>
<div class="line"><a name="l00222"></a><span class="lineno">  222</span>&#160;  <span class="comment">//  - 0 will be -INF</span></div>
<div class="line"><a name="l00223"></a><span class="lineno">  223</span>&#160;  <span class="comment">//  - +INF will be +INF</span></div>
<div class="line"><a name="l00224"></a><span class="lineno">  224</span>&#160;  <span class="keywordflow">return</span> pselect(iszero_mask, cst_minus_inf,</div>
<div class="line"><a name="l00225"></a><span class="lineno">  225</span>&#160;                              por(pselect(pos_inf_mask,cst_pos_inf,x), invalid_mask));</div>
<div class="line"><a name="l00226"></a><span class="lineno">  226</span>&#160;}</div>
<div class="line"><a name="l00227"></a><span class="lineno">  227</span>&#160; </div>
<div class="line"><a name="l00228"></a><span class="lineno">  228</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00229"></a><span class="lineno">  229</span>&#160;EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS</div>
<div class="line"><a name="l00230"></a><span class="lineno">  230</span>&#160;Packet plog_float(<span class="keyword">const</span> Packet _x)</div>
<div class="line"><a name="l00231"></a><span class="lineno">  231</span>&#160;{</div>
<div class="line"><a name="l00232"></a><span class="lineno">  232</span>&#160;  <span class="keywordflow">return</span> plog_impl_float&lt;Packet, <span class="comment">/* base2 */</span> <span class="keyword">false</span>&gt;(_x);</div>
<div class="line"><a name="l00233"></a><span class="lineno">  233</span>&#160;}</div>
<div class="line"><a name="l00234"></a><span class="lineno">  234</span>&#160; </div>
<div class="line"><a name="l00235"></a><span class="lineno">  235</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00236"></a><span class="lineno">  236</span>&#160;EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS</div>
<div class="line"><a name="l00237"></a><span class="lineno">  237</span>&#160;Packet plog2_float(<span class="keyword">const</span> Packet _x)</div>
<div class="line"><a name="l00238"></a><span class="lineno">  238</span>&#160;{</div>
<div class="line"><a name="l00239"></a><span class="lineno">  239</span>&#160;  <span class="keywordflow">return</span> plog_impl_float&lt;Packet, <span class="comment">/* base2 */</span> <span class="keyword">true</span>&gt;(_x);</div>
<div class="line"><a name="l00240"></a><span class="lineno">  240</span>&#160;}</div>
<div class="line"><a name="l00241"></a><span class="lineno">  241</span>&#160; </div>
<div class="line"><a name="l00242"></a><span class="lineno">  242</span>&#160;<span class="comment">/* Returns the base e (2.718...) or base 2 logarithm of x.</span></div>
<div class="line"><a name="l00243"></a><span class="lineno">  243</span>&#160;<span class="comment"> * The argument is separated into its exponent and fractional parts.</span></div>
<div class="line"><a name="l00244"></a><span class="lineno">  244</span>&#160;<span class="comment"> * The logarithm of the fraction in the interval [sqrt(1/2), sqrt(2)],</span></div>
<div class="line"><a name="l00245"></a><span class="lineno">  245</span>&#160;<span class="comment"> * is approximated by</span></div>
<div class="line"><a name="l00246"></a><span class="lineno">  246</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00247"></a><span class="lineno">  247</span>&#160;<span class="comment"> *     log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).</span></div>
<div class="line"><a name="l00248"></a><span class="lineno">  248</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00249"></a><span class="lineno">  249</span>&#160;<span class="comment"> * for more detail see: http://www.netlib.org/cephes/</span></div>
<div class="line"><a name="l00250"></a><span class="lineno">  250</span>&#160;<span class="comment"> */</span></div>
<div class="line"><a name="l00251"></a><span class="lineno">  251</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> Packet, <span class="keywordtype">bool</span> base2&gt;</div>
<div class="line"><a name="l00252"></a><span class="lineno">  252</span>&#160;EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS</div>
<div class="line"><a name="l00253"></a><span class="lineno">  253</span>&#160;Packet plog_impl_double(<span class="keyword">const</span> Packet _x)</div>
<div class="line"><a name="l00254"></a><span class="lineno">  254</span>&#160;{</div>
<div class="line"><a name="l00255"></a><span class="lineno">  255</span>&#160;  Packet x = _x;</div>
<div class="line"><a name="l00256"></a><span class="lineno">  256</span>&#160; </div>
<div class="line"><a name="l00257"></a><span class="lineno">  257</span>&#160;  <span class="keyword">const</span> Packet cst_1              = pset1&lt;Packet&gt;(1.0);</div>
<div class="line"><a name="l00258"></a><span class="lineno">  258</span>&#160;  <span class="keyword">const</span> Packet cst_neg_half       = pset1&lt;Packet&gt;(-0.5);</div>
<div class="line"><a name="l00259"></a><span class="lineno">  259</span>&#160;  <span class="keyword">const</span> Packet cst_minus_inf      = pset1frombits&lt;Packet&gt;( <span class="keyword">static_cast&lt;</span>uint64_t<span class="keyword">&gt;</span>(0xfff0000000000000ull));</div>
<div class="line"><a name="l00260"></a><span class="lineno">  260</span>&#160;  <span class="keyword">const</span> Packet cst_pos_inf        = pset1frombits&lt;Packet&gt;( <span class="keyword">static_cast&lt;</span>uint64_t<span class="keyword">&gt;</span>(0x7ff0000000000000ull));</div>
<div class="line"><a name="l00261"></a><span class="lineno">  261</span>&#160; </div>
<div class="line"><a name="l00262"></a><span class="lineno">  262</span>&#160; </div>
<div class="line"><a name="l00263"></a><span class="lineno">  263</span>&#160; <span class="comment">// Polynomial Coefficients for log(1+x) = x - x**2/2 + x**3 P(x)/Q(x)</span></div>
<div class="line"><a name="l00264"></a><span class="lineno">  264</span>&#160; <span class="comment">//                             1/sqrt(2) &lt;= x &lt; sqrt(2)</span></div>
<div class="line"><a name="l00265"></a><span class="lineno">  265</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_SQRTHF = pset1&lt;Packet&gt;(0.70710678118654752440E0);</div>
<div class="line"><a name="l00266"></a><span class="lineno">  266</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_log_p0 = pset1&lt;Packet&gt;(1.01875663804580931796E-4);</div>
<div class="line"><a name="l00267"></a><span class="lineno">  267</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_log_p1 = pset1&lt;Packet&gt;(4.97494994976747001425E-1);</div>
<div class="line"><a name="l00268"></a><span class="lineno">  268</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_log_p2 = pset1&lt;Packet&gt;(4.70579119878881725854E0);</div>
<div class="line"><a name="l00269"></a><span class="lineno">  269</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_log_p3 = pset1&lt;Packet&gt;(1.44989225341610930846E1);</div>
<div class="line"><a name="l00270"></a><span class="lineno">  270</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_log_p4 = pset1&lt;Packet&gt;(1.79368678507819816313E1);</div>
<div class="line"><a name="l00271"></a><span class="lineno">  271</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_log_p5 = pset1&lt;Packet&gt;(7.70838733755885391666E0);</div>
<div class="line"><a name="l00272"></a><span class="lineno">  272</span>&#160; </div>
<div class="line"><a name="l00273"></a><span class="lineno">  273</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_log_q0 = pset1&lt;Packet&gt;(1.0);</div>
<div class="line"><a name="l00274"></a><span class="lineno">  274</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_log_q1 = pset1&lt;Packet&gt;(1.12873587189167450590E1);</div>
<div class="line"><a name="l00275"></a><span class="lineno">  275</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_log_q2 = pset1&lt;Packet&gt;(4.52279145837532221105E1);</div>
<div class="line"><a name="l00276"></a><span class="lineno">  276</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_log_q3 = pset1&lt;Packet&gt;(8.29875266912776603211E1);</div>
<div class="line"><a name="l00277"></a><span class="lineno">  277</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_log_q4 = pset1&lt;Packet&gt;(7.11544750618563894466E1);</div>
<div class="line"><a name="l00278"></a><span class="lineno">  278</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_log_q5 = pset1&lt;Packet&gt;(2.31251620126765340583E1);</div>
<div class="line"><a name="l00279"></a><span class="lineno">  279</span>&#160; </div>
<div class="line"><a name="l00280"></a><span class="lineno">  280</span>&#160;  Packet e;</div>
<div class="line"><a name="l00281"></a><span class="lineno">  281</span>&#160;  <span class="comment">// extract significant in the range [0.5,1) and exponent</span></div>
<div class="line"><a name="l00282"></a><span class="lineno">  282</span>&#160;  x = pfrexp(x,e);</div>
<div class="line"><a name="l00283"></a><span class="lineno">  283</span>&#160;  </div>
<div class="line"><a name="l00284"></a><span class="lineno">  284</span>&#160;  <span class="comment">// Shift the inputs from the range [0.5,1) to [sqrt(1/2),sqrt(2))</span></div>
<div class="line"><a name="l00285"></a><span class="lineno">  285</span>&#160;  <span class="comment">// and shift by -1. The values are then centered around 0, which improves</span></div>
<div class="line"><a name="l00286"></a><span class="lineno">  286</span>&#160;  <span class="comment">// the stability of the polynomial evaluation.</span></div>
<div class="line"><a name="l00287"></a><span class="lineno">  287</span>&#160;  <span class="comment">//   if( x &lt; SQRTHF ) {</span></div>
<div class="line"><a name="l00288"></a><span class="lineno">  288</span>&#160;  <span class="comment">//     e -= 1;</span></div>
<div class="line"><a name="l00289"></a><span class="lineno">  289</span>&#160;  <span class="comment">//     x = x + x - 1.0;</span></div>
<div class="line"><a name="l00290"></a><span class="lineno">  290</span>&#160;  <span class="comment">//   } else { x = x - 1.0; }</span></div>
<div class="line"><a name="l00291"></a><span class="lineno">  291</span>&#160;  Packet mask = pcmp_lt(x, cst_cephes_SQRTHF);</div>
<div class="line"><a name="l00292"></a><span class="lineno">  292</span>&#160;  Packet tmp = pand(x, mask);</div>
<div class="line"><a name="l00293"></a><span class="lineno">  293</span>&#160;  x = psub(x, cst_1);</div>
<div class="line"><a name="l00294"></a><span class="lineno">  294</span>&#160;  e = psub(e, pand(cst_1, mask));</div>
<div class="line"><a name="l00295"></a><span class="lineno">  295</span>&#160;  x = padd(x, tmp);</div>
<div class="line"><a name="l00296"></a><span class="lineno">  296</span>&#160; </div>
<div class="line"><a name="l00297"></a><span class="lineno">  297</span>&#160;  Packet x2 = pmul(x, x);</div>
<div class="line"><a name="l00298"></a><span class="lineno">  298</span>&#160;  Packet x3 = pmul(x2, x);</div>
<div class="line"><a name="l00299"></a><span class="lineno">  299</span>&#160; </div>
<div class="line"><a name="l00300"></a><span class="lineno">  300</span>&#160;  <span class="comment">// Evaluate the polynomial approximant , probably to improve instruction-level parallelism.</span></div>
<div class="line"><a name="l00301"></a><span class="lineno">  301</span>&#160;  <span class="comment">// y = x - 0.5*x^2 + x^3 * polevl( x, P, 5 ) / p1evl( x, Q, 5 ) );</span></div>
<div class="line"><a name="l00302"></a><span class="lineno">  302</span>&#160;  Packet y, y1, y_;</div>
<div class="line"><a name="l00303"></a><span class="lineno">  303</span>&#160;  y  = pmadd(cst_cephes_log_p0, x, cst_cephes_log_p1);</div>
<div class="line"><a name="l00304"></a><span class="lineno">  304</span>&#160;  y1 = pmadd(cst_cephes_log_p3, x, cst_cephes_log_p4);</div>
<div class="line"><a name="l00305"></a><span class="lineno">  305</span>&#160;  y  = pmadd(y, x, cst_cephes_log_p2);</div>
<div class="line"><a name="l00306"></a><span class="lineno">  306</span>&#160;  y1 = pmadd(y1, x, cst_cephes_log_p5);</div>
<div class="line"><a name="l00307"></a><span class="lineno">  307</span>&#160;  y_ = pmadd(y, x3, y1);</div>
<div class="line"><a name="l00308"></a><span class="lineno">  308</span>&#160; </div>
<div class="line"><a name="l00309"></a><span class="lineno">  309</span>&#160;  y  = pmadd(cst_cephes_log_q0, x, cst_cephes_log_q1);</div>
<div class="line"><a name="l00310"></a><span class="lineno">  310</span>&#160;  y1 = pmadd(cst_cephes_log_q3, x, cst_cephes_log_q4);</div>
<div class="line"><a name="l00311"></a><span class="lineno">  311</span>&#160;  y  = pmadd(y, x, cst_cephes_log_q2);</div>
<div class="line"><a name="l00312"></a><span class="lineno">  312</span>&#160;  y1 = pmadd(y1, x, cst_cephes_log_q5);</div>
<div class="line"><a name="l00313"></a><span class="lineno">  313</span>&#160;  y  = pmadd(y, x3, y1);</div>
<div class="line"><a name="l00314"></a><span class="lineno">  314</span>&#160; </div>
<div class="line"><a name="l00315"></a><span class="lineno">  315</span>&#160;  y_ = pmul(y_, x3);</div>
<div class="line"><a name="l00316"></a><span class="lineno">  316</span>&#160;  y  = pdiv(y_, y);</div>
<div class="line"><a name="l00317"></a><span class="lineno">  317</span>&#160; </div>
<div class="line"><a name="l00318"></a><span class="lineno">  318</span>&#160;  y = pmadd(cst_neg_half, x2, y);</div>
<div class="line"><a name="l00319"></a><span class="lineno">  319</span>&#160;  x = padd(x, y);</div>
<div class="line"><a name="l00320"></a><span class="lineno">  320</span>&#160; </div>
<div class="line"><a name="l00321"></a><span class="lineno">  321</span>&#160;  <span class="comment">// Add the logarithm of the exponent back to the result of the interpolation.</span></div>
<div class="line"><a name="l00322"></a><span class="lineno">  322</span>&#160;  <span class="keywordflow">if</span> (base2) {</div>
<div class="line"><a name="l00323"></a><span class="lineno">  323</span>&#160;    <span class="keyword">const</span> Packet cst_log2e = pset1&lt;Packet&gt;(<span class="keyword">static_cast&lt;</span><span class="keywordtype">double</span><span class="keyword">&gt;</span>(EIGEN_LOG2E));</div>
<div class="line"><a name="l00324"></a><span class="lineno">  324</span>&#160;    x = pmadd(x, cst_log2e, e);</div>
<div class="line"><a name="l00325"></a><span class="lineno">  325</span>&#160;  } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00326"></a><span class="lineno">  326</span>&#160;    <span class="keyword">const</span> Packet cst_ln2 = pset1&lt;Packet&gt;(<span class="keyword">static_cast&lt;</span><span class="keywordtype">double</span><span class="keyword">&gt;</span>(EIGEN_LN2));</div>
<div class="line"><a name="l00327"></a><span class="lineno">  327</span>&#160;    x = pmadd(e, cst_ln2, x);</div>
<div class="line"><a name="l00328"></a><span class="lineno">  328</span>&#160;  }</div>
<div class="line"><a name="l00329"></a><span class="lineno">  329</span>&#160; </div>
<div class="line"><a name="l00330"></a><span class="lineno">  330</span>&#160;  Packet invalid_mask = pcmp_lt_or_nan(_x, pzero(_x));</div>
<div class="line"><a name="l00331"></a><span class="lineno">  331</span>&#160;  Packet iszero_mask  = pcmp_eq(_x,pzero(_x));</div>
<div class="line"><a name="l00332"></a><span class="lineno">  332</span>&#160;  Packet pos_inf_mask = pcmp_eq(_x,cst_pos_inf);</div>
<div class="line"><a name="l00333"></a><span class="lineno">  333</span>&#160;  <span class="comment">// Filter out invalid inputs, i.e.:</span></div>
<div class="line"><a name="l00334"></a><span class="lineno">  334</span>&#160;  <span class="comment">//  - negative arg will be NAN</span></div>
<div class="line"><a name="l00335"></a><span class="lineno">  335</span>&#160;  <span class="comment">//  - 0 will be -INF</span></div>
<div class="line"><a name="l00336"></a><span class="lineno">  336</span>&#160;  <span class="comment">//  - +INF will be +INF</span></div>
<div class="line"><a name="l00337"></a><span class="lineno">  337</span>&#160;  <span class="keywordflow">return</span> pselect(iszero_mask, cst_minus_inf,</div>
<div class="line"><a name="l00338"></a><span class="lineno">  338</span>&#160;                              por(pselect(pos_inf_mask,cst_pos_inf,x), invalid_mask));</div>
<div class="line"><a name="l00339"></a><span class="lineno">  339</span>&#160;}</div>
<div class="line"><a name="l00340"></a><span class="lineno">  340</span>&#160; </div>
<div class="line"><a name="l00341"></a><span class="lineno">  341</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00342"></a><span class="lineno">  342</span>&#160;EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS</div>
<div class="line"><a name="l00343"></a><span class="lineno">  343</span>&#160;Packet plog_double(<span class="keyword">const</span> Packet _x)</div>
<div class="line"><a name="l00344"></a><span class="lineno">  344</span>&#160;{</div>
<div class="line"><a name="l00345"></a><span class="lineno">  345</span>&#160;  <span class="keywordflow">return</span> plog_impl_double&lt;Packet, <span class="comment">/* base2 */</span> <span class="keyword">false</span>&gt;(_x);</div>
<div class="line"><a name="l00346"></a><span class="lineno">  346</span>&#160;}</div>
<div class="line"><a name="l00347"></a><span class="lineno">  347</span>&#160; </div>
<div class="line"><a name="l00348"></a><span class="lineno">  348</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00349"></a><span class="lineno">  349</span>&#160;EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS</div>
<div class="line"><a name="l00350"></a><span class="lineno">  350</span>&#160;Packet plog2_double(<span class="keyword">const</span> Packet _x)</div>
<div class="line"><a name="l00351"></a><span class="lineno">  351</span>&#160;{</div>
<div class="line"><a name="l00352"></a><span class="lineno">  352</span>&#160;  <span class="keywordflow">return</span> plog_impl_double&lt;Packet, <span class="comment">/* base2 */</span> <span class="keyword">true</span>&gt;(_x);</div>
<div class="line"><a name="l00353"></a><span class="lineno">  353</span>&#160;}</div>
<div class="line"><a name="l00354"></a><span class="lineno">  354</span>&#160; </div>
<div class="line"><a name="l00358"></a><span class="lineno">  358</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00359"></a><span class="lineno">  359</span>&#160;Packet generic_plog1p(<span class="keyword">const</span> Packet&amp; x)</div>
<div class="line"><a name="l00360"></a><span class="lineno">  360</span>&#160;{</div>
<div class="line"><a name="l00361"></a><span class="lineno">  361</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::type ScalarType;</div>
<div class="line"><a name="l00362"></a><span class="lineno">  362</span>&#160;  <span class="keyword">const</span> Packet one = pset1&lt;Packet&gt;(ScalarType(1));</div>
<div class="line"><a name="l00363"></a><span class="lineno">  363</span>&#160;  Packet xp1 = padd(x, one);</div>
<div class="line"><a name="l00364"></a><span class="lineno">  364</span>&#160;  Packet small_mask = pcmp_eq(xp1, one);</div>
<div class="line"><a name="l00365"></a><span class="lineno">  365</span>&#160;  Packet log1 = plog(xp1);</div>
<div class="line"><a name="l00366"></a><span class="lineno">  366</span>&#160;  Packet inf_mask = pcmp_eq(xp1, log1);</div>
<div class="line"><a name="l00367"></a><span class="lineno">  367</span>&#160;  Packet log_large = pmul(x, pdiv(log1, psub(xp1, one)));</div>
<div class="line"><a name="l00368"></a><span class="lineno">  368</span>&#160;  <span class="keywordflow">return</span> pselect(por(small_mask, inf_mask), x, log_large);</div>
<div class="line"><a name="l00369"></a><span class="lineno">  369</span>&#160;}</div>
<div class="line"><a name="l00370"></a><span class="lineno">  370</span>&#160; </div>
<div class="line"><a name="l00374"></a><span class="lineno">  374</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00375"></a><span class="lineno">  375</span>&#160;Packet generic_expm1(<span class="keyword">const</span> Packet&amp; x)</div>
<div class="line"><a name="l00376"></a><span class="lineno">  376</span>&#160;{</div>
<div class="line"><a name="l00377"></a><span class="lineno">  377</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::type ScalarType;</div>
<div class="line"><a name="l00378"></a><span class="lineno">  378</span>&#160;  <span class="keyword">const</span> Packet one = pset1&lt;Packet&gt;(ScalarType(1));</div>
<div class="line"><a name="l00379"></a><span class="lineno">  379</span>&#160;  <span class="keyword">const</span> Packet neg_one = pset1&lt;Packet&gt;(ScalarType(-1));</div>
<div class="line"><a name="l00380"></a><span class="lineno">  380</span>&#160;  Packet u = pexp(x);</div>
<div class="line"><a name="l00381"></a><span class="lineno">  381</span>&#160;  Packet one_mask = pcmp_eq(u, one);</div>
<div class="line"><a name="l00382"></a><span class="lineno">  382</span>&#160;  Packet u_minus_one = psub(u, one);</div>
<div class="line"><a name="l00383"></a><span class="lineno">  383</span>&#160;  Packet neg_one_mask = pcmp_eq(u_minus_one, neg_one);</div>
<div class="line"><a name="l00384"></a><span class="lineno">  384</span>&#160;  Packet logu = plog(u);</div>
<div class="line"><a name="l00385"></a><span class="lineno">  385</span>&#160;  <span class="comment">// The following comparison is to catch the case where</span></div>
<div class="line"><a name="l00386"></a><span class="lineno">  386</span>&#160;  <span class="comment">// exp(x) = +inf. It is written in this way to avoid having</span></div>
<div class="line"><a name="l00387"></a><span class="lineno">  387</span>&#160;  <span class="comment">// to form the constant +inf, which depends on the packet</span></div>
<div class="line"><a name="l00388"></a><span class="lineno">  388</span>&#160;  <span class="comment">// type.</span></div>
<div class="line"><a name="l00389"></a><span class="lineno">  389</span>&#160;  Packet pos_inf_mask = pcmp_eq(logu, u);</div>
<div class="line"><a name="l00390"></a><span class="lineno">  390</span>&#160;  Packet <a class="code" href="namespaceEigen.html#ae7cb2544e4e745bc0067fe793e3f2f81">expm1</a> = pmul(u_minus_one, pdiv(x, logu));</div>
<div class="line"><a name="l00391"></a><span class="lineno">  391</span>&#160;  <a class="code" href="namespaceEigen.html#ae7cb2544e4e745bc0067fe793e3f2f81">expm1</a> = pselect(pos_inf_mask, u, <a class="code" href="namespaceEigen.html#ae7cb2544e4e745bc0067fe793e3f2f81">expm1</a>);</div>
<div class="line"><a name="l00392"></a><span class="lineno">  392</span>&#160;  <span class="keywordflow">return</span> pselect(one_mask,</div>
<div class="line"><a name="l00393"></a><span class="lineno">  393</span>&#160;                 x,</div>
<div class="line"><a name="l00394"></a><span class="lineno">  394</span>&#160;                 pselect(neg_one_mask,</div>
<div class="line"><a name="l00395"></a><span class="lineno">  395</span>&#160;                         neg_one,</div>
<div class="line"><a name="l00396"></a><span class="lineno">  396</span>&#160;                         <a class="code" href="namespaceEigen.html#ae7cb2544e4e745bc0067fe793e3f2f81">expm1</a>));</div>
<div class="line"><a name="l00397"></a><span class="lineno">  397</span>&#160;}</div>
<div class="line"><a name="l00398"></a><span class="lineno">  398</span>&#160; </div>
<div class="line"><a name="l00399"></a><span class="lineno">  399</span>&#160; </div>
<div class="line"><a name="l00400"></a><span class="lineno">  400</span>&#160;<span class="comment">// Exponential function. Works by writing &quot;x = m*log(2) + r&quot; where</span></div>
<div class="line"><a name="l00401"></a><span class="lineno">  401</span>&#160;<span class="comment">// &quot;m = floor(x/log(2)+1/2)&quot; and &quot;r&quot; is the remainder. The result is then</span></div>
<div class="line"><a name="l00402"></a><span class="lineno">  402</span>&#160;<span class="comment">// &quot;exp(x) = 2^m*exp(r)&quot; where exp(r) is in the range [-1,1).</span></div>
<div class="line"><a name="l00403"></a><span class="lineno">  403</span>&#160;<span class="comment">// exp(r) is computed using a 6th order minimax polynomial approximation.</span></div>
<div class="line"><a name="l00404"></a><span class="lineno">  404</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00405"></a><span class="lineno">  405</span>&#160;EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS</div>
<div class="line"><a name="l00406"></a><span class="lineno">  406</span>&#160;Packet pexp_float(<span class="keyword">const</span> Packet _x)</div>
<div class="line"><a name="l00407"></a><span class="lineno">  407</span>&#160;{</div>
<div class="line"><a name="l00408"></a><span class="lineno">  408</span>&#160;  <span class="keyword">const</span> Packet cst_zero   = pset1&lt;Packet&gt;(0.0f);</div>
<div class="line"><a name="l00409"></a><span class="lineno">  409</span>&#160;  <span class="keyword">const</span> Packet cst_one    = pset1&lt;Packet&gt;(1.0f);</div>
<div class="line"><a name="l00410"></a><span class="lineno">  410</span>&#160;  <span class="keyword">const</span> Packet cst_half   = pset1&lt;Packet&gt;(0.5f);</div>
<div class="line"><a name="l00411"></a><span class="lineno">  411</span>&#160;  <span class="keyword">const</span> Packet cst_exp_hi = pset1&lt;Packet&gt;(88.723f);</div>
<div class="line"><a name="l00412"></a><span class="lineno">  412</span>&#160;  <span class="keyword">const</span> Packet cst_exp_lo = pset1&lt;Packet&gt;(-104.f);</div>
<div class="line"><a name="l00413"></a><span class="lineno">  413</span>&#160; </div>
<div class="line"><a name="l00414"></a><span class="lineno">  414</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_LOG2EF = pset1&lt;Packet&gt;(1.44269504088896341f);</div>
<div class="line"><a name="l00415"></a><span class="lineno">  415</span>&#160;  <span class="keyword">const</span> Packet cst_p2 = pset1&lt;Packet&gt;(0.49999988079071044921875f);</div>
<div class="line"><a name="l00416"></a><span class="lineno">  416</span>&#160;  <span class="keyword">const</span> Packet cst_p3 = pset1&lt;Packet&gt;(0.16666518151760101318359375f);</div>
<div class="line"><a name="l00417"></a><span class="lineno">  417</span>&#160;  <span class="keyword">const</span> Packet cst_p4 = pset1&lt;Packet&gt;(4.166965186595916748046875e-2f);</div>
<div class="line"><a name="l00418"></a><span class="lineno">  418</span>&#160;  <span class="keyword">const</span> Packet cst_p5 = pset1&lt;Packet&gt;(8.36894474923610687255859375e-3f);</div>
<div class="line"><a name="l00419"></a><span class="lineno">  419</span>&#160;  <span class="keyword">const</span> Packet cst_p6 = pset1&lt;Packet&gt;(1.37449637986719608306884765625e-3f);</div>
<div class="line"><a name="l00420"></a><span class="lineno">  420</span>&#160; </div>
<div class="line"><a name="l00421"></a><span class="lineno">  421</span>&#160;  <span class="comment">// Clamp x.</span></div>
<div class="line"><a name="l00422"></a><span class="lineno">  422</span>&#160;  Packet zero_mask = pcmp_lt(_x, cst_exp_lo);</div>
<div class="line"><a name="l00423"></a><span class="lineno">  423</span>&#160;  Packet x = pmin(_x, cst_exp_hi);</div>
<div class="line"><a name="l00424"></a><span class="lineno">  424</span>&#160; </div>
<div class="line"><a name="l00425"></a><span class="lineno">  425</span>&#160;  <span class="comment">// Express exp(x) as exp(m*ln(2) + r), start by extracting</span></div>
<div class="line"><a name="l00426"></a><span class="lineno">  426</span>&#160;  <span class="comment">// m = floor(x/ln(2) + 0.5).</span></div>
<div class="line"><a name="l00427"></a><span class="lineno">  427</span>&#160;  Packet m = pfloor(pmadd(x, cst_cephes_LOG2EF, cst_half));</div>
<div class="line"><a name="l00428"></a><span class="lineno">  428</span>&#160; </div>
<div class="line"><a name="l00429"></a><span class="lineno">  429</span>&#160;  <span class="comment">// Get r = x - m*ln(2). If no FMA instructions are available, m*ln(2) is</span></div>
<div class="line"><a name="l00430"></a><span class="lineno">  430</span>&#160;  <span class="comment">// subtracted out in two parts, m*C1+m*C2 = m*ln(2), to avoid accumulating</span></div>
<div class="line"><a name="l00431"></a><span class="lineno">  431</span>&#160;  <span class="comment">// truncation errors.</span></div>
<div class="line"><a name="l00432"></a><span class="lineno">  432</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_exp_C1 = pset1&lt;Packet&gt;(-0.693359375f);</div>
<div class="line"><a name="l00433"></a><span class="lineno">  433</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_exp_C2 = pset1&lt;Packet&gt;(2.12194440e-4f);</div>
<div class="line"><a name="l00434"></a><span class="lineno">  434</span>&#160;  Packet r = pmadd(m, cst_cephes_exp_C1, x);</div>
<div class="line"><a name="l00435"></a><span class="lineno">  435</span>&#160;  r = pmadd(m, cst_cephes_exp_C2, r);</div>
<div class="line"><a name="l00436"></a><span class="lineno">  436</span>&#160; </div>
<div class="line"><a name="l00437"></a><span class="lineno">  437</span>&#160;  <span class="comment">// Evaluate the 6th order polynomial approximation to exp(r)</span></div>
<div class="line"><a name="l00438"></a><span class="lineno">  438</span>&#160;  <span class="comment">// with r in the interval [-ln(2)/2;ln(2)/2].</span></div>
<div class="line"><a name="l00439"></a><span class="lineno">  439</span>&#160;  <span class="keyword">const</span> Packet r2 = pmul(r, r);</div>
<div class="line"><a name="l00440"></a><span class="lineno">  440</span>&#160;  Packet p_even = pmadd(r2, cst_p6, cst_p4);</div>
<div class="line"><a name="l00441"></a><span class="lineno">  441</span>&#160;  <span class="keyword">const</span> Packet p_odd = pmadd(r2, cst_p5, cst_p3);</div>
<div class="line"><a name="l00442"></a><span class="lineno">  442</span>&#160;  p_even = pmadd(r2, p_even, cst_p2);</div>
<div class="line"><a name="l00443"></a><span class="lineno">  443</span>&#160;  <span class="keyword">const</span> Packet p_low = padd(r, cst_one);</div>
<div class="line"><a name="l00444"></a><span class="lineno">  444</span>&#160;  Packet y = pmadd(r, p_odd, p_even);</div>
<div class="line"><a name="l00445"></a><span class="lineno">  445</span>&#160;  y = pmadd(r2, y, p_low);</div>
<div class="line"><a name="l00446"></a><span class="lineno">  446</span>&#160; </div>
<div class="line"><a name="l00447"></a><span class="lineno">  447</span>&#160;  <span class="comment">// Return 2^m * exp(r).</span></div>
<div class="line"><a name="l00448"></a><span class="lineno">  448</span>&#160;  <span class="comment">// TODO: replace pldexp with faster implementation since y in [-1, 1).</span></div>
<div class="line"><a name="l00449"></a><span class="lineno">  449</span>&#160;  <span class="keywordflow">return</span> pselect(zero_mask, cst_zero, pmax(pldexp(y,m), _x));</div>
<div class="line"><a name="l00450"></a><span class="lineno">  450</span>&#160;}</div>
<div class="line"><a name="l00451"></a><span class="lineno">  451</span>&#160; </div>
<div class="line"><a name="l00452"></a><span class="lineno">  452</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00453"></a><span class="lineno">  453</span>&#160;EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS</div>
<div class="line"><a name="l00454"></a><span class="lineno">  454</span>&#160;Packet pexp_double(<span class="keyword">const</span> Packet _x)</div>
<div class="line"><a name="l00455"></a><span class="lineno">  455</span>&#160;{</div>
<div class="line"><a name="l00456"></a><span class="lineno">  456</span>&#160;  Packet x = _x;</div>
<div class="line"><a name="l00457"></a><span class="lineno">  457</span>&#160;  <span class="keyword">const</span> Packet cst_zero = pset1&lt;Packet&gt;(0.0);</div>
<div class="line"><a name="l00458"></a><span class="lineno">  458</span>&#160;  <span class="keyword">const</span> Packet cst_1 = pset1&lt;Packet&gt;(1.0);</div>
<div class="line"><a name="l00459"></a><span class="lineno">  459</span>&#160;  <span class="keyword">const</span> Packet cst_2 = pset1&lt;Packet&gt;(2.0);</div>
<div class="line"><a name="l00460"></a><span class="lineno">  460</span>&#160;  <span class="keyword">const</span> Packet cst_half = pset1&lt;Packet&gt;(0.5);</div>
<div class="line"><a name="l00461"></a><span class="lineno">  461</span>&#160; </div>
<div class="line"><a name="l00462"></a><span class="lineno">  462</span>&#160;  <span class="keyword">const</span> Packet cst_exp_hi = pset1&lt;Packet&gt;(709.784);</div>
<div class="line"><a name="l00463"></a><span class="lineno">  463</span>&#160;  <span class="keyword">const</span> Packet cst_exp_lo = pset1&lt;Packet&gt;(-709.784);</div>
<div class="line"><a name="l00464"></a><span class="lineno">  464</span>&#160; </div>
<div class="line"><a name="l00465"></a><span class="lineno">  465</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_LOG2EF = pset1&lt;Packet&gt;(1.4426950408889634073599);</div>
<div class="line"><a name="l00466"></a><span class="lineno">  466</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_exp_p0 = pset1&lt;Packet&gt;(1.26177193074810590878e-4);</div>
<div class="line"><a name="l00467"></a><span class="lineno">  467</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_exp_p1 = pset1&lt;Packet&gt;(3.02994407707441961300e-2);</div>
<div class="line"><a name="l00468"></a><span class="lineno">  468</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_exp_p2 = pset1&lt;Packet&gt;(9.99999999999999999910e-1);</div>
<div class="line"><a name="l00469"></a><span class="lineno">  469</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_exp_q0 = pset1&lt;Packet&gt;(3.00198505138664455042e-6);</div>
<div class="line"><a name="l00470"></a><span class="lineno">  470</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_exp_q1 = pset1&lt;Packet&gt;(2.52448340349684104192e-3);</div>
<div class="line"><a name="l00471"></a><span class="lineno">  471</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_exp_q2 = pset1&lt;Packet&gt;(2.27265548208155028766e-1);</div>
<div class="line"><a name="l00472"></a><span class="lineno">  472</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_exp_q3 = pset1&lt;Packet&gt;(2.00000000000000000009e0);</div>
<div class="line"><a name="l00473"></a><span class="lineno">  473</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_exp_C1 = pset1&lt;Packet&gt;(0.693145751953125);</div>
<div class="line"><a name="l00474"></a><span class="lineno">  474</span>&#160;  <span class="keyword">const</span> Packet cst_cephes_exp_C2 = pset1&lt;Packet&gt;(1.42860682030941723212e-6);</div>
<div class="line"><a name="l00475"></a><span class="lineno">  475</span>&#160; </div>
<div class="line"><a name="l00476"></a><span class="lineno">  476</span>&#160;  Packet tmp, fx;</div>
<div class="line"><a name="l00477"></a><span class="lineno">  477</span>&#160; </div>
<div class="line"><a name="l00478"></a><span class="lineno">  478</span>&#160;  <span class="comment">// clamp x</span></div>
<div class="line"><a name="l00479"></a><span class="lineno">  479</span>&#160;  Packet zero_mask = pcmp_lt(_x, cst_exp_lo);</div>
<div class="line"><a name="l00480"></a><span class="lineno">  480</span>&#160;  x = pmin(x, cst_exp_hi);</div>
<div class="line"><a name="l00481"></a><span class="lineno">  481</span>&#160;  <span class="comment">// Express exp(x) as exp(g + n*log(2)).</span></div>
<div class="line"><a name="l00482"></a><span class="lineno">  482</span>&#160;  fx = pmadd(cst_cephes_LOG2EF, x, cst_half);</div>
<div class="line"><a name="l00483"></a><span class="lineno">  483</span>&#160; </div>
<div class="line"><a name="l00484"></a><span class="lineno">  484</span>&#160;  <span class="comment">// Get the integer modulus of log(2), i.e. the &quot;n&quot; described above.</span></div>
<div class="line"><a name="l00485"></a><span class="lineno">  485</span>&#160;  fx = pfloor(fx);</div>
<div class="line"><a name="l00486"></a><span class="lineno">  486</span>&#160; </div>
<div class="line"><a name="l00487"></a><span class="lineno">  487</span>&#160;  <span class="comment">// Get the remainder modulo log(2), i.e. the &quot;g&quot; described above. Subtract</span></div>
<div class="line"><a name="l00488"></a><span class="lineno">  488</span>&#160;  <span class="comment">// n*log(2) out in two steps, i.e. n*C1 + n*C2, C1+C2=log2 to get the last</span></div>
<div class="line"><a name="l00489"></a><span class="lineno">  489</span>&#160;  <span class="comment">// digits right.</span></div>
<div class="line"><a name="l00490"></a><span class="lineno">  490</span>&#160;  tmp = pmul(fx, cst_cephes_exp_C1);</div>
<div class="line"><a name="l00491"></a><span class="lineno">  491</span>&#160;  Packet z = pmul(fx, cst_cephes_exp_C2);</div>
<div class="line"><a name="l00492"></a><span class="lineno">  492</span>&#160;  x = psub(x, tmp);</div>
<div class="line"><a name="l00493"></a><span class="lineno">  493</span>&#160;  x = psub(x, z);</div>
<div class="line"><a name="l00494"></a><span class="lineno">  494</span>&#160; </div>
<div class="line"><a name="l00495"></a><span class="lineno">  495</span>&#160;  Packet x2 = pmul(x, x);</div>
<div class="line"><a name="l00496"></a><span class="lineno">  496</span>&#160; </div>
<div class="line"><a name="l00497"></a><span class="lineno">  497</span>&#160;  <span class="comment">// Evaluate the numerator polynomial of the rational interpolant.</span></div>
<div class="line"><a name="l00498"></a><span class="lineno">  498</span>&#160;  Packet px = cst_cephes_exp_p0;</div>
<div class="line"><a name="l00499"></a><span class="lineno">  499</span>&#160;  px = pmadd(px, x2, cst_cephes_exp_p1);</div>
<div class="line"><a name="l00500"></a><span class="lineno">  500</span>&#160;  px = pmadd(px, x2, cst_cephes_exp_p2);</div>
<div class="line"><a name="l00501"></a><span class="lineno">  501</span>&#160;  px = pmul(px, x);</div>
<div class="line"><a name="l00502"></a><span class="lineno">  502</span>&#160; </div>
<div class="line"><a name="l00503"></a><span class="lineno">  503</span>&#160;  <span class="comment">// Evaluate the denominator polynomial of the rational interpolant.</span></div>
<div class="line"><a name="l00504"></a><span class="lineno">  504</span>&#160;  Packet qx = cst_cephes_exp_q0;</div>
<div class="line"><a name="l00505"></a><span class="lineno">  505</span>&#160;  qx = pmadd(qx, x2, cst_cephes_exp_q1);</div>
<div class="line"><a name="l00506"></a><span class="lineno">  506</span>&#160;  qx = pmadd(qx, x2, cst_cephes_exp_q2);</div>
<div class="line"><a name="l00507"></a><span class="lineno">  507</span>&#160;  qx = pmadd(qx, x2, cst_cephes_exp_q3);</div>
<div class="line"><a name="l00508"></a><span class="lineno">  508</span>&#160; </div>
<div class="line"><a name="l00509"></a><span class="lineno">  509</span>&#160;  <span class="comment">// I don&#39;t really get this bit, copied from the SSE2 routines, so...</span></div>
<div class="line"><a name="l00510"></a><span class="lineno">  510</span>&#160;  <span class="comment">// TODO(gonnet): Figure out what is going on here, perhaps find a better</span></div>
<div class="line"><a name="l00511"></a><span class="lineno">  511</span>&#160;  <span class="comment">// rational interpolant?</span></div>
<div class="line"><a name="l00512"></a><span class="lineno">  512</span>&#160;  x = pdiv(px, psub(qx, px));</div>
<div class="line"><a name="l00513"></a><span class="lineno">  513</span>&#160;  x = pmadd(cst_2, x, cst_1);</div>
<div class="line"><a name="l00514"></a><span class="lineno">  514</span>&#160; </div>
<div class="line"><a name="l00515"></a><span class="lineno">  515</span>&#160;  <span class="comment">// Construct the result 2^n * exp(g) = e * x. The max is used to catch</span></div>
<div class="line"><a name="l00516"></a><span class="lineno">  516</span>&#160;  <span class="comment">// non-finite values in the input.</span></div>
<div class="line"><a name="l00517"></a><span class="lineno">  517</span>&#160;  <span class="comment">// TODO: replace pldexp with faster implementation since x in [-1, 1).</span></div>
<div class="line"><a name="l00518"></a><span class="lineno">  518</span>&#160;  <span class="keywordflow">return</span> pselect(zero_mask, cst_zero, pmax(pldexp(x,fx), _x));</div>
<div class="line"><a name="l00519"></a><span class="lineno">  519</span>&#160;}</div>
<div class="line"><a name="l00520"></a><span class="lineno">  520</span>&#160; </div>
<div class="line"><a name="l00521"></a><span class="lineno">  521</span>&#160;<span class="comment">// The following code is inspired by the following stack-overflow answer:</span></div>
<div class="line"><a name="l00522"></a><span class="lineno">  522</span>&#160;<span class="comment">//   https://stackoverflow.com/questions/30463616/payne-hanek-algorithm-implementation-in-c/30465751#30465751</span></div>
<div class="line"><a name="l00523"></a><span class="lineno">  523</span>&#160;<span class="comment">// It has been largely optimized:</span></div>
<div class="line"><a name="l00524"></a><span class="lineno">  524</span>&#160;<span class="comment">//  - By-pass calls to frexp.</span></div>
<div class="line"><a name="l00525"></a><span class="lineno">  525</span>&#160;<span class="comment">//  - Aligned loads of required 96 bits of 2/pi. This is accomplished by</span></div>
<div class="line"><a name="l00526"></a><span class="lineno">  526</span>&#160;<span class="comment">//    (1) balancing the mantissa and exponent to the required bits of 2/pi are</span></div>
<div class="line"><a name="l00527"></a><span class="lineno">  527</span>&#160;<span class="comment">//    aligned on 8-bits, and (2) replicating the storage of the bits of 2/pi.</span></div>
<div class="line"><a name="l00528"></a><span class="lineno">  528</span>&#160;<span class="comment">//  - Avoid a branch in rounding and extraction of the remaining fractional part.</span></div>
<div class="line"><a name="l00529"></a><span class="lineno">  529</span>&#160;<span class="comment">// Overall, I measured a speed up higher than x2 on x86-64.</span></div>
<div class="line"><a name="l00530"></a><span class="lineno">  530</span>&#160;<span class="keyword">inline</span> <span class="keywordtype">float</span> trig_reduce_huge (<span class="keywordtype">float</span> xf, Eigen::numext::int32_t *quadrant)</div>
<div class="line"><a name="l00531"></a><span class="lineno">  531</span>&#160;{</div>
<div class="line"><a name="l00532"></a><span class="lineno">  532</span>&#160;  <span class="keyword">using</span> Eigen::numext::int32_t;</div>
<div class="line"><a name="l00533"></a><span class="lineno">  533</span>&#160;  <span class="keyword">using</span> Eigen::numext::uint32_t;</div>
<div class="line"><a name="l00534"></a><span class="lineno">  534</span>&#160;  <span class="keyword">using</span> Eigen::numext::int64_t;</div>
<div class="line"><a name="l00535"></a><span class="lineno">  535</span>&#160;  <span class="keyword">using</span> Eigen::numext::uint64_t;</div>
<div class="line"><a name="l00536"></a><span class="lineno">  536</span>&#160; </div>
<div class="line"><a name="l00537"></a><span class="lineno">  537</span>&#160;  <span class="keyword">const</span> <span class="keywordtype">double</span> pio2_62 = 3.4061215800865545e-19;    <span class="comment">// pi/2 * 2^-62</span></div>
<div class="line"><a name="l00538"></a><span class="lineno">  538</span>&#160;  <span class="keyword">const</span> uint64_t zero_dot_five = uint64_t(1) &lt;&lt; 61; <span class="comment">// 0.5 in 2.62-bit fixed-point format</span></div>
<div class="line"><a name="l00539"></a><span class="lineno">  539</span>&#160; </div>
<div class="line"><a name="l00540"></a><span class="lineno">  540</span>&#160;  <span class="comment">// 192 bits of 2/pi for Payne-Hanek reduction</span></div>
<div class="line"><a name="l00541"></a><span class="lineno">  541</span>&#160;  <span class="comment">// Bits are introduced by packet of 8 to enable aligned reads.</span></div>
<div class="line"><a name="l00542"></a><span class="lineno">  542</span>&#160;  <span class="keyword">static</span> <span class="keyword">const</span> uint32_t two_over_pi [] = </div>
<div class="line"><a name="l00543"></a><span class="lineno">  543</span>&#160;  {</div>
<div class="line"><a name="l00544"></a><span class="lineno">  544</span>&#160;    0x00000028, 0x000028be, 0x0028be60, 0x28be60db,</div>
<div class="line"><a name="l00545"></a><span class="lineno">  545</span>&#160;    0xbe60db93, 0x60db9391, 0xdb939105, 0x9391054a,</div>
<div class="line"><a name="l00546"></a><span class="lineno">  546</span>&#160;    0x91054a7f, 0x054a7f09, 0x4a7f09d5, 0x7f09d5f4,</div>
<div class="line"><a name="l00547"></a><span class="lineno">  547</span>&#160;    0x09d5f47d, 0xd5f47d4d, 0xf47d4d37, 0x7d4d3770,</div>
<div class="line"><a name="l00548"></a><span class="lineno">  548</span>&#160;    0x4d377036, 0x377036d8, 0x7036d8a5, 0x36d8a566,</div>
<div class="line"><a name="l00549"></a><span class="lineno">  549</span>&#160;    0xd8a5664f, 0xa5664f10, 0x664f10e4, 0x4f10e410,</div>
<div class="line"><a name="l00550"></a><span class="lineno">  550</span>&#160;    0x10e41000, 0xe4100000</div>
<div class="line"><a name="l00551"></a><span class="lineno">  551</span>&#160;  };</div>
<div class="line"><a name="l00552"></a><span class="lineno">  552</span>&#160;  </div>
<div class="line"><a name="l00553"></a><span class="lineno">  553</span>&#160;  uint32_t xi = numext::bit_cast&lt;uint32_t&gt;(xf);</div>
<div class="line"><a name="l00554"></a><span class="lineno">  554</span>&#160;  <span class="comment">// Below, -118 = -126 + 8.</span></div>
<div class="line"><a name="l00555"></a><span class="lineno">  555</span>&#160;  <span class="comment">//   -126 is to get the exponent,</span></div>
<div class="line"><a name="l00556"></a><span class="lineno">  556</span>&#160;  <span class="comment">//   +8 is to enable alignment of 2/pi&#39;s bits on 8 bits.</span></div>
<div class="line"><a name="l00557"></a><span class="lineno">  557</span>&#160;  <span class="comment">// This is possible because the fractional part of x as only 24 meaningful bits.</span></div>
<div class="line"><a name="l00558"></a><span class="lineno">  558</span>&#160;  uint32_t e = (xi &gt;&gt; 23) - 118;</div>
<div class="line"><a name="l00559"></a><span class="lineno">  559</span>&#160;  <span class="comment">// Extract the mantissa and shift it to align it wrt the exponent</span></div>
<div class="line"><a name="l00560"></a><span class="lineno">  560</span>&#160;  xi = ((xi &amp; 0x007fffffu)| 0x00800000u) &lt;&lt; (e &amp; 0x7);</div>
<div class="line"><a name="l00561"></a><span class="lineno">  561</span>&#160; </div>
<div class="line"><a name="l00562"></a><span class="lineno">  562</span>&#160;  uint32_t i = e &gt;&gt; 3;</div>
<div class="line"><a name="l00563"></a><span class="lineno">  563</span>&#160;  uint32_t twoopi_1  = two_over_pi[i-1];</div>
<div class="line"><a name="l00564"></a><span class="lineno">  564</span>&#160;  uint32_t twoopi_2  = two_over_pi[i+3];</div>
<div class="line"><a name="l00565"></a><span class="lineno">  565</span>&#160;  uint32_t twoopi_3  = two_over_pi[i+7];</div>
<div class="line"><a name="l00566"></a><span class="lineno">  566</span>&#160; </div>
<div class="line"><a name="l00567"></a><span class="lineno">  567</span>&#160;  <span class="comment">// Compute x * 2/pi in 2.62-bit fixed-point format.</span></div>
<div class="line"><a name="l00568"></a><span class="lineno">  568</span>&#160;  uint64_t p;</div>
<div class="line"><a name="l00569"></a><span class="lineno">  569</span>&#160;  p = uint64_t(xi) * twoopi_3;</div>
<div class="line"><a name="l00570"></a><span class="lineno">  570</span>&#160;  p = uint64_t(xi) * twoopi_2 + (p &gt;&gt; 32);</div>
<div class="line"><a name="l00571"></a><span class="lineno">  571</span>&#160;  p = (uint64_t(xi * twoopi_1) &lt;&lt; 32) + p;</div>
<div class="line"><a name="l00572"></a><span class="lineno">  572</span>&#160; </div>
<div class="line"><a name="l00573"></a><span class="lineno">  573</span>&#160;  <span class="comment">// Round to nearest: add 0.5 and extract integral part.</span></div>
<div class="line"><a name="l00574"></a><span class="lineno">  574</span>&#160;  uint64_t q = (p + zero_dot_five) &gt;&gt; 62;</div>
<div class="line"><a name="l00575"></a><span class="lineno">  575</span>&#160;  *quadrant = int(q);</div>
<div class="line"><a name="l00576"></a><span class="lineno">  576</span>&#160;  <span class="comment">// Now it remains to compute &quot;r = x - q*pi/2&quot; with high accuracy,</span></div>
<div class="line"><a name="l00577"></a><span class="lineno">  577</span>&#160;  <span class="comment">// since we have p=x/(pi/2) with high accuracy, we can more efficiently compute r as:</span></div>
<div class="line"><a name="l00578"></a><span class="lineno">  578</span>&#160;  <span class="comment">//   r = (p-q)*pi/2,</span></div>
<div class="line"><a name="l00579"></a><span class="lineno">  579</span>&#160;  <span class="comment">// where the product can be be carried out with sufficient accuracy using double precision.</span></div>
<div class="line"><a name="l00580"></a><span class="lineno">  580</span>&#160;  p -= q&lt;&lt;62;</div>
<div class="line"><a name="l00581"></a><span class="lineno">  581</span>&#160;  <span class="keywordflow">return</span> float(<span class="keywordtype">double</span>(int64_t(p)) * pio2_62);</div>
<div class="line"><a name="l00582"></a><span class="lineno">  582</span>&#160;}</div>
<div class="line"><a name="l00583"></a><span class="lineno">  583</span>&#160; </div>
<div class="line"><a name="l00584"></a><span class="lineno">  584</span>&#160;<span class="keyword">template</span>&lt;<span class="keywordtype">bool</span> ComputeSine,<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00585"></a><span class="lineno">  585</span>&#160;EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS</div>
<div class="line"><a name="l00586"></a><span class="lineno">  586</span>&#160;<span class="preprocessor">#if EIGEN_COMP_GNUC_STRICT</span></div>
<div class="line"><a name="l00587"></a><span class="lineno">  587</span>&#160;__attribute__((optimize(<span class="stringliteral">&quot;-fno-unsafe-math-optimizations&quot;</span>)))</div>
<div class="line"><a name="l00588"></a><span class="lineno">  588</span>&#160;<span class="preprocessor">#endif</span></div>
<div class="line"><a name="l00589"></a><span class="lineno">  589</span>&#160;Packet psincos_float(<span class="keyword">const</span> Packet&amp; _x)</div>
<div class="line"><a name="l00590"></a><span class="lineno">  590</span>&#160;{</div>
<div class="line"><a name="l00591"></a><span class="lineno">  591</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::integer_packet PacketI;</div>
<div class="line"><a name="l00592"></a><span class="lineno">  592</span>&#160; </div>
<div class="line"><a name="l00593"></a><span class="lineno">  593</span>&#160;  <span class="keyword">const</span> Packet  cst_2oPI            = pset1&lt;Packet&gt;(0.636619746685028076171875f); <span class="comment">// 2/PI</span></div>
<div class="line"><a name="l00594"></a><span class="lineno">  594</span>&#160;  <span class="keyword">const</span> Packet  cst_rounding_magic  = pset1&lt;Packet&gt;(12582912); <span class="comment">// 2^23 for rounding</span></div>
<div class="line"><a name="l00595"></a><span class="lineno">  595</span>&#160;  <span class="keyword">const</span> PacketI csti_1              = pset1&lt;PacketI&gt;(1);</div>
<div class="line"><a name="l00596"></a><span class="lineno">  596</span>&#160;  <span class="keyword">const</span> Packet  cst_sign_mask       = pset1frombits&lt;Packet&gt;(<span class="keyword">static_cast&lt;</span>Eigen::numext::uint32_t<span class="keyword">&gt;</span>(0x80000000u));</div>
<div class="line"><a name="l00597"></a><span class="lineno">  597</span>&#160; </div>
<div class="line"><a name="l00598"></a><span class="lineno">  598</span>&#160;  Packet x = pabs(_x);</div>
<div class="line"><a name="l00599"></a><span class="lineno">  599</span>&#160; </div>
<div class="line"><a name="l00600"></a><span class="lineno">  600</span>&#160;  <span class="comment">// Scale x by 2/Pi to find x&#39;s octant.</span></div>
<div class="line"><a name="l00601"></a><span class="lineno">  601</span>&#160;  Packet y = pmul(x, cst_2oPI);</div>
<div class="line"><a name="l00602"></a><span class="lineno">  602</span>&#160; </div>
<div class="line"><a name="l00603"></a><span class="lineno">  603</span>&#160;  <span class="comment">// Rounding trick to find nearest integer:</span></div>
<div class="line"><a name="l00604"></a><span class="lineno">  604</span>&#160;  Packet y_round = padd(y, cst_rounding_magic);</div>
<div class="line"><a name="l00605"></a><span class="lineno">  605</span>&#160;  EIGEN_OPTIMIZATION_BARRIER(y_round)</div>
<div class="line"><a name="l00606"></a><span class="lineno">  606</span>&#160;  PacketI y_int = preinterpret&lt;PacketI&gt;(y_round); <span class="comment">// last 23 digits represent integer (if abs(x)&lt;2^24)</span></div>
<div class="line"><a name="l00607"></a><span class="lineno">  607</span>&#160;  y = psub(y_round, cst_rounding_magic); <span class="comment">// nearest integer to x * (2/pi)</span></div>
<div class="line"><a name="l00608"></a><span class="lineno">  608</span>&#160; </div>
<div class="line"><a name="l00609"></a><span class="lineno">  609</span>&#160;  <span class="comment">// Subtract y * Pi/2 to reduce x to the interval -Pi/4 &lt;= x &lt;= +Pi/4</span></div>
<div class="line"><a name="l00610"></a><span class="lineno">  610</span>&#160;  <span class="comment">// using &quot;Extended precision modular arithmetic&quot;</span></div>
<div class="line"><a name="l00611"></a><span class="lineno">  611</span>&#160;<span class="preprocessor">  #if defined(EIGEN_HAS_SINGLE_INSTRUCTION_MADD)</span></div>
<div class="line"><a name="l00612"></a><span class="lineno">  612</span>&#160;  <span class="comment">// This version requires true FMA for high accuracy</span></div>
<div class="line"><a name="l00613"></a><span class="lineno">  613</span>&#160;  <span class="comment">// It provides a max error of 1ULP up to (with absolute_error &lt; 5.9605e-08):</span></div>
<div class="line"><a name="l00614"></a><span class="lineno">  614</span>&#160;  <span class="keyword">const</span> <span class="keywordtype">float</span> huge_th = ComputeSine ? 117435.992f : 71476.0625f;</div>
<div class="line"><a name="l00615"></a><span class="lineno">  615</span>&#160;  x = pmadd(y, pset1&lt;Packet&gt;(-1.57079601287841796875f), x);</div>
<div class="line"><a name="l00616"></a><span class="lineno">  616</span>&#160;  x = pmadd(y, pset1&lt;Packet&gt;(-3.1391647326017846353352069854736328125e-07f), x);</div>
<div class="line"><a name="l00617"></a><span class="lineno">  617</span>&#160;  x = pmadd(y, pset1&lt;Packet&gt;(-5.390302529957764765544681040410068817436695098876953125e-15f), x);</div>
<div class="line"><a name="l00618"></a><span class="lineno">  618</span>&#160;<span class="preprocessor">  #else</span></div>
<div class="line"><a name="l00619"></a><span class="lineno">  619</span>&#160;  <span class="comment">// Without true FMA, the previous set of coefficients maintain 1ULP accuracy</span></div>
<div class="line"><a name="l00620"></a><span class="lineno">  620</span>&#160;  <span class="comment">// up to x&lt;15.7 (for sin), but accuracy is immediately lost for x&gt;15.7.</span></div>
<div class="line"><a name="l00621"></a><span class="lineno">  621</span>&#160;  <span class="comment">// We thus use one more iteration to maintain 2ULPs up to reasonably large inputs.</span></div>
<div class="line"><a name="l00622"></a><span class="lineno">  622</span>&#160; </div>
<div class="line"><a name="l00623"></a><span class="lineno">  623</span>&#160;  <span class="comment">// The following set of coefficients maintain 1ULP up to 9.43 and 14.16 for sin and cos respectively.</span></div>
<div class="line"><a name="l00624"></a><span class="lineno">  624</span>&#160;  <span class="comment">// and 2 ULP up to:</span></div>
<div class="line"><a name="l00625"></a><span class="lineno">  625</span>&#160;  <span class="keyword">const</span> <span class="keywordtype">float</span> huge_th = ComputeSine ? 25966.f : 18838.f;</div>
<div class="line"><a name="l00626"></a><span class="lineno">  626</span>&#160;  x = pmadd(y, pset1&lt;Packet&gt;(-1.5703125), x); <span class="comment">// = 0xbfc90000</span></div>
<div class="line"><a name="l00627"></a><span class="lineno">  627</span>&#160;  EIGEN_OPTIMIZATION_BARRIER(x)</div>
<div class="line"><a name="l00628"></a><span class="lineno">  628</span>&#160;  x = pmadd(y, pset1&lt;Packet&gt;(-0.000483989715576171875), x); <span class="comment">// = 0xb9fdc000</span></div>
<div class="line"><a name="l00629"></a><span class="lineno">  629</span>&#160;  EIGEN_OPTIMIZATION_BARRIER(x)</div>
<div class="line"><a name="l00630"></a><span class="lineno">  630</span>&#160;  x = pmadd(y, pset1&lt;Packet&gt;(1.62865035235881805419921875e-07), x); <span class="comment">// = 0x342ee000</span></div>
<div class="line"><a name="l00631"></a><span class="lineno">  631</span>&#160;  x = pmadd(y, pset1&lt;Packet&gt;(5.5644315544167710640977020375430583953857421875e-11), x); <span class="comment">// = 0x2e74b9ee</span></div>
<div class="line"><a name="l00632"></a><span class="lineno">  632</span>&#160; </div>
<div class="line"><a name="l00633"></a><span class="lineno">  633</span>&#160;  <span class="comment">// For the record, the following set of coefficients maintain 2ULP up</span></div>
<div class="line"><a name="l00634"></a><span class="lineno">  634</span>&#160;  <span class="comment">// to a slightly larger range:</span></div>
<div class="line"><a name="l00635"></a><span class="lineno">  635</span>&#160;  <span class="comment">// const float huge_th = ComputeSine ? 51981.f : 39086.125f;</span></div>
<div class="line"><a name="l00636"></a><span class="lineno">  636</span>&#160;  <span class="comment">// but it slightly fails to maintain 1ULP for two values of sin below pi.</span></div>
<div class="line"><a name="l00637"></a><span class="lineno">  637</span>&#160;  <span class="comment">// x = pmadd(y, pset1&lt;Packet&gt;(-3.140625/2.), x);</span></div>
<div class="line"><a name="l00638"></a><span class="lineno">  638</span>&#160;  <span class="comment">// x = pmadd(y, pset1&lt;Packet&gt;(-0.00048351287841796875), x);</span></div>
<div class="line"><a name="l00639"></a><span class="lineno">  639</span>&#160;  <span class="comment">// x = pmadd(y, pset1&lt;Packet&gt;(-3.13855707645416259765625e-07), x);</span></div>
<div class="line"><a name="l00640"></a><span class="lineno">  640</span>&#160;  <span class="comment">// x = pmadd(y, pset1&lt;Packet&gt;(-6.0771006282767103812147979624569416046142578125e-11), x);</span></div>
<div class="line"><a name="l00641"></a><span class="lineno">  641</span>&#160; </div>
<div class="line"><a name="l00642"></a><span class="lineno">  642</span>&#160;  <span class="comment">// For the record, with only 3 iterations it is possible to maintain</span></div>
<div class="line"><a name="l00643"></a><span class="lineno">  643</span>&#160;  <span class="comment">// 1 ULP up to 3PI (maybe more) and 2ULP up to 255.</span></div>
<div class="line"><a name="l00644"></a><span class="lineno">  644</span>&#160;  <span class="comment">// The coefficients are: 0xbfc90f80, 0xb7354480, 0x2e74b9ee</span></div>
<div class="line"><a name="l00645"></a><span class="lineno">  645</span>&#160;<span class="preprocessor">  #endif</span></div>
<div class="line"><a name="l00646"></a><span class="lineno">  646</span>&#160; </div>
<div class="line"><a name="l00647"></a><span class="lineno">  647</span>&#160;  <span class="keywordflow">if</span>(predux_any(pcmp_le(pset1&lt;Packet&gt;(huge_th),pabs(_x))))</div>
<div class="line"><a name="l00648"></a><span class="lineno">  648</span>&#160;  {</div>
<div class="line"><a name="l00649"></a><span class="lineno">  649</span>&#160;    <span class="keyword">const</span> <span class="keywordtype">int</span> PacketSize = unpacket_traits&lt;Packet&gt;::size;</div>
<div class="line"><a name="l00650"></a><span class="lineno">  650</span>&#160;    EIGEN_ALIGN_TO_BOUNDARY(<span class="keyword">sizeof</span>(Packet)) <span class="keywordtype">float</span> vals[PacketSize];</div>
<div class="line"><a name="l00651"></a><span class="lineno">  651</span>&#160;    EIGEN_ALIGN_TO_BOUNDARY(<span class="keyword">sizeof</span>(Packet)) <span class="keywordtype">float</span> x_cpy[PacketSize];</div>
<div class="line"><a name="l00652"></a><span class="lineno">  652</span>&#160;    EIGEN_ALIGN_TO_BOUNDARY(<span class="keyword">sizeof</span>(Packet)) Eigen::numext::int32_t y_int2[PacketSize];</div>
<div class="line"><a name="l00653"></a><span class="lineno">  653</span>&#160;    pstoreu(vals, pabs(_x));</div>
<div class="line"><a name="l00654"></a><span class="lineno">  654</span>&#160;    pstoreu(x_cpy, x);</div>
<div class="line"><a name="l00655"></a><span class="lineno">  655</span>&#160;    pstoreu(y_int2, y_int);</div>
<div class="line"><a name="l00656"></a><span class="lineno">  656</span>&#160;    <span class="keywordflow">for</span>(<span class="keywordtype">int</span> k=0; k&lt;PacketSize;++k)</div>
<div class="line"><a name="l00657"></a><span class="lineno">  657</span>&#160;    {</div>
<div class="line"><a name="l00658"></a><span class="lineno">  658</span>&#160;      <span class="keywordtype">float</span> val = vals[k];</div>
<div class="line"><a name="l00659"></a><span class="lineno">  659</span>&#160;      <span class="keywordflow">if</span>(val&gt;=huge_th &amp;&amp; (numext::isfinite)(val))</div>
<div class="line"><a name="l00660"></a><span class="lineno">  660</span>&#160;        x_cpy[k] = trig_reduce_huge(val,&amp;y_int2[k]);</div>
<div class="line"><a name="l00661"></a><span class="lineno">  661</span>&#160;    }</div>
<div class="line"><a name="l00662"></a><span class="lineno">  662</span>&#160;    x = ploadu&lt;Packet&gt;(x_cpy);</div>
<div class="line"><a name="l00663"></a><span class="lineno">  663</span>&#160;    y_int = ploadu&lt;PacketI&gt;(y_int2);</div>
<div class="line"><a name="l00664"></a><span class="lineno">  664</span>&#160;  }</div>
<div class="line"><a name="l00665"></a><span class="lineno">  665</span>&#160; </div>
<div class="line"><a name="l00666"></a><span class="lineno">  666</span>&#160;  <span class="comment">// Compute the sign to apply to the polynomial.</span></div>
<div class="line"><a name="l00667"></a><span class="lineno">  667</span>&#160;  <span class="comment">// sin: sign = second_bit(y_int) xor signbit(_x)</span></div>
<div class="line"><a name="l00668"></a><span class="lineno">  668</span>&#160;  <span class="comment">// cos: sign = second_bit(y_int+1)</span></div>
<div class="line"><a name="l00669"></a><span class="lineno">  669</span>&#160;  Packet sign_bit = ComputeSine ? pxor(_x, preinterpret&lt;Packet&gt;(plogical_shift_left&lt;30&gt;(y_int)))</div>
<div class="line"><a name="l00670"></a><span class="lineno">  670</span>&#160;                                : preinterpret&lt;Packet&gt;(plogical_shift_left&lt;30&gt;(padd(y_int,csti_1)));</div>
<div class="line"><a name="l00671"></a><span class="lineno">  671</span>&#160;  sign_bit = pand(sign_bit, cst_sign_mask); <span class="comment">// clear all but left most bit</span></div>
<div class="line"><a name="l00672"></a><span class="lineno">  672</span>&#160; </div>
<div class="line"><a name="l00673"></a><span class="lineno">  673</span>&#160;  <span class="comment">// Get the polynomial selection mask from the second bit of y_int</span></div>
<div class="line"><a name="l00674"></a><span class="lineno">  674</span>&#160;  <span class="comment">// We&#39;ll calculate both (sin and cos) polynomials and then select from the two.</span></div>
<div class="line"><a name="l00675"></a><span class="lineno">  675</span>&#160;  Packet poly_mask = preinterpret&lt;Packet&gt;(pcmp_eq(pand(y_int, csti_1), pzero(y_int)));</div>
<div class="line"><a name="l00676"></a><span class="lineno">  676</span>&#160; </div>
<div class="line"><a name="l00677"></a><span class="lineno">  677</span>&#160;  Packet x2 = pmul(x,x);</div>
<div class="line"><a name="l00678"></a><span class="lineno">  678</span>&#160; </div>
<div class="line"><a name="l00679"></a><span class="lineno">  679</span>&#160;  <span class="comment">// Evaluate the cos(x) polynomial. (-Pi/4 &lt;= x &lt;= Pi/4)</span></div>
<div class="line"><a name="l00680"></a><span class="lineno">  680</span>&#160;  Packet y1 =        pset1&lt;Packet&gt;(2.4372266125283204019069671630859375e-05f);</div>
<div class="line"><a name="l00681"></a><span class="lineno">  681</span>&#160;  y1 = pmadd(y1, x2, pset1&lt;Packet&gt;(-0.00138865201734006404876708984375f     ));</div>
<div class="line"><a name="l00682"></a><span class="lineno">  682</span>&#160;  y1 = pmadd(y1, x2, pset1&lt;Packet&gt;(0.041666619479656219482421875f           ));</div>
<div class="line"><a name="l00683"></a><span class="lineno">  683</span>&#160;  y1 = pmadd(y1, x2, pset1&lt;Packet&gt;(-0.5f));</div>
<div class="line"><a name="l00684"></a><span class="lineno">  684</span>&#160;  y1 = pmadd(y1, x2, pset1&lt;Packet&gt;(1.f));</div>
<div class="line"><a name="l00685"></a><span class="lineno">  685</span>&#160; </div>
<div class="line"><a name="l00686"></a><span class="lineno">  686</span>&#160;  <span class="comment">// Evaluate the sin(x) polynomial. (Pi/4 &lt;= x &lt;= Pi/4)</span></div>
<div class="line"><a name="l00687"></a><span class="lineno">  687</span>&#160;  <span class="comment">// octave/matlab code to compute those coefficients:</span></div>
<div class="line"><a name="l00688"></a><span class="lineno">  688</span>&#160;  <span class="comment">//    x = (0:0.0001:pi/4)&#39;;</span></div>
<div class="line"><a name="l00689"></a><span class="lineno">  689</span>&#160;  <span class="comment">//    A = [x.^3 x.^5 x.^7];</span></div>
<div class="line"><a name="l00690"></a><span class="lineno">  690</span>&#160;  <span class="comment">//    w = ((1.-(x/(pi/4)).^2).^5)*2000+1;         # weights trading relative accuracy</span></div>
<div class="line"><a name="l00691"></a><span class="lineno">  691</span>&#160;  <span class="comment">//    c = (A&#39;*diag(w)*A)\&zwj;(A&#39;*diag(w)*(sin(x)-x)); # weighted LS, linear coeff forced to 1</span></div>
<div class="line"><a name="l00692"></a><span class="lineno">  692</span>&#160;  <span class="comment">//    printf(&#39;%.64f\n %.64f\n%.64f\n&#39;, c(3), c(2), c(1))</span></div>
<div class="line"><a name="l00693"></a><span class="lineno">  693</span>&#160;  <span class="comment">//</span></div>
<div class="line"><a name="l00694"></a><span class="lineno">  694</span>&#160;  Packet y2 =        pset1&lt;Packet&gt;(-0.0001959234114083702898469196984621021329076029360294342041015625f);</div>
<div class="line"><a name="l00695"></a><span class="lineno">  695</span>&#160;  y2 = pmadd(y2, x2, pset1&lt;Packet&gt;( 0.0083326873655616851693794799871284340042620897293090820312500000f));</div>
<div class="line"><a name="l00696"></a><span class="lineno">  696</span>&#160;  y2 = pmadd(y2, x2, pset1&lt;Packet&gt;(-0.1666666203982298255503735617821803316473960876464843750000000000f));</div>
<div class="line"><a name="l00697"></a><span class="lineno">  697</span>&#160;  y2 = pmul(y2, x2);</div>
<div class="line"><a name="l00698"></a><span class="lineno">  698</span>&#160;  y2 = pmadd(y2, x, x);</div>
<div class="line"><a name="l00699"></a><span class="lineno">  699</span>&#160; </div>
<div class="line"><a name="l00700"></a><span class="lineno">  700</span>&#160;  <span class="comment">// Select the correct result from the two polynomials.</span></div>
<div class="line"><a name="l00701"></a><span class="lineno">  701</span>&#160;  y = ComputeSine ? pselect(poly_mask,y2,y1)</div>
<div class="line"><a name="l00702"></a><span class="lineno">  702</span>&#160;                  : pselect(poly_mask,y1,y2);</div>
<div class="line"><a name="l00703"></a><span class="lineno">  703</span>&#160; </div>
<div class="line"><a name="l00704"></a><span class="lineno">  704</span>&#160;  <span class="comment">// Update the sign and filter huge inputs</span></div>
<div class="line"><a name="l00705"></a><span class="lineno">  705</span>&#160;  <span class="keywordflow">return</span> pxor(y, sign_bit);</div>
<div class="line"><a name="l00706"></a><span class="lineno">  706</span>&#160;}</div>
<div class="line"><a name="l00707"></a><span class="lineno">  707</span>&#160; </div>
<div class="line"><a name="l00708"></a><span class="lineno">  708</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00709"></a><span class="lineno">  709</span>&#160;EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS</div>
<div class="line"><a name="l00710"></a><span class="lineno">  710</span>&#160;Packet psin_float(<span class="keyword">const</span> Packet&amp; x)</div>
<div class="line"><a name="l00711"></a><span class="lineno">  711</span>&#160;{</div>
<div class="line"><a name="l00712"></a><span class="lineno">  712</span>&#160;  <span class="keywordflow">return</span> psincos_float&lt;true&gt;(x);</div>
<div class="line"><a name="l00713"></a><span class="lineno">  713</span>&#160;}</div>
<div class="line"><a name="l00714"></a><span class="lineno">  714</span>&#160; </div>
<div class="line"><a name="l00715"></a><span class="lineno">  715</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00716"></a><span class="lineno">  716</span>&#160;EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS</div>
<div class="line"><a name="l00717"></a><span class="lineno">  717</span>&#160;Packet pcos_float(<span class="keyword">const</span> Packet&amp; x)</div>
<div class="line"><a name="l00718"></a><span class="lineno">  718</span>&#160;{</div>
<div class="line"><a name="l00719"></a><span class="lineno">  719</span>&#160;  <span class="keywordflow">return</span> psincos_float&lt;false&gt;(x);</div>
<div class="line"><a name="l00720"></a><span class="lineno">  720</span>&#160;}</div>
<div class="line"><a name="l00721"></a><span class="lineno">  721</span>&#160; </div>
<div class="line"><a name="l00722"></a><span class="lineno">  722</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00723"></a><span class="lineno">  723</span>&#160;EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS</div>
<div class="line"><a name="l00724"></a><span class="lineno">  724</span>&#160;Packet pdiv_complex(<span class="keyword">const</span> Packet&amp; x, <span class="keyword">const</span> Packet&amp; y) {</div>
<div class="line"><a name="l00725"></a><span class="lineno">  725</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::as_real RealPacket;</div>
<div class="line"><a name="l00726"></a><span class="lineno">  726</span>&#160;  <span class="comment">// In the following we annotate the code for the case where the inputs</span></div>
<div class="line"><a name="l00727"></a><span class="lineno">  727</span>&#160;  <span class="comment">// are a pair length-2 SIMD vectors representing a single pair of complex</span></div>
<div class="line"><a name="l00728"></a><span class="lineno">  728</span>&#160;  <span class="comment">// numbers x = a + i*b, y = c + i*d.</span></div>
<div class="line"><a name="l00729"></a><span class="lineno">  729</span>&#160;  <span class="keyword">const</span> RealPacket y_abs = pabs(y.v);  <span class="comment">// |c|, |d|</span></div>
<div class="line"><a name="l00730"></a><span class="lineno">  730</span>&#160;  <span class="keyword">const</span> RealPacket y_abs_flip = pcplxflip(Packet(y_abs)).v; <span class="comment">// |d|, |c|</span></div>
<div class="line"><a name="l00731"></a><span class="lineno">  731</span>&#160;  <span class="keyword">const</span> RealPacket y_max = pmax(y_abs, y_abs_flip); <span class="comment">// max(|c|, |d|), max(|c|, |d|)</span></div>
<div class="line"><a name="l00732"></a><span class="lineno">  732</span>&#160;  <span class="keyword">const</span> RealPacket y_scaled = pdiv(y.v, y_max);  <span class="comment">// c / max(|c|, |d|), d / max(|c|, |d|)</span></div>
<div class="line"><a name="l00733"></a><span class="lineno">  733</span>&#160;  <span class="comment">// Compute scaled denominator.</span></div>
<div class="line"><a name="l00734"></a><span class="lineno">  734</span>&#160;  <span class="keyword">const</span> RealPacket y_scaled_sq = pmul(y_scaled, y_scaled); <span class="comment">// c&#39;**2, d&#39;**2</span></div>
<div class="line"><a name="l00735"></a><span class="lineno">  735</span>&#160;  <span class="keyword">const</span> RealPacket denom = padd(y_scaled_sq, pcplxflip(Packet(y_scaled_sq)).v);</div>
<div class="line"><a name="l00736"></a><span class="lineno">  736</span>&#160;  Packet result_scaled = pmul(x, pconj(Packet(y_scaled)));  <span class="comment">// a * c&#39; + b * d&#39;, -a * d + b * c</span></div>
<div class="line"><a name="l00737"></a><span class="lineno">  737</span>&#160;  <span class="comment">// Divide elementwise by denom.</span></div>
<div class="line"><a name="l00738"></a><span class="lineno">  738</span>&#160;  result_scaled = Packet(pdiv(result_scaled.v, denom));</div>
<div class="line"><a name="l00739"></a><span class="lineno">  739</span>&#160;  <span class="comment">// Rescale result</span></div>
<div class="line"><a name="l00740"></a><span class="lineno">  740</span>&#160;  <span class="keywordflow">return</span> Packet(pdiv(result_scaled.v, y_max));</div>
<div class="line"><a name="l00741"></a><span class="lineno">  741</span>&#160;}</div>
<div class="line"><a name="l00742"></a><span class="lineno">  742</span>&#160; </div>
<div class="line"><a name="l00743"></a><span class="lineno">  743</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00744"></a><span class="lineno">  744</span>&#160;EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS</div>
<div class="line"><a name="l00745"></a><span class="lineno">  745</span>&#160;Packet psqrt_complex(<span class="keyword">const</span> Packet&amp; a) {</div>
<div class="line"><a name="l00746"></a><span class="lineno">  746</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::type Scalar;</div>
<div class="line"><a name="l00747"></a><span class="lineno">  747</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> Scalar::value_type RealScalar;</div>
<div class="line"><a name="l00748"></a><span class="lineno">  748</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::as_real RealPacket;</div>
<div class="line"><a name="l00749"></a><span class="lineno">  749</span>&#160; </div>
<div class="line"><a name="l00750"></a><span class="lineno">  750</span>&#160;  <span class="comment">// Computes the principal sqrt of the complex numbers in the input.</span></div>
<div class="line"><a name="l00751"></a><span class="lineno">  751</span>&#160;  <span class="comment">//</span></div>
<div class="line"><a name="l00752"></a><span class="lineno">  752</span>&#160;  <span class="comment">// For example, for packets containing 2 complex numbers stored in interleaved format</span></div>
<div class="line"><a name="l00753"></a><span class="lineno">  753</span>&#160;  <span class="comment">//    a = [a0, a1] = [x0, y0, x1, y1],</span></div>
<div class="line"><a name="l00754"></a><span class="lineno">  754</span>&#160;  <span class="comment">// where x0 = real(a0), y0 = imag(a0) etc., this function returns</span></div>
<div class="line"><a name="l00755"></a><span class="lineno">  755</span>&#160;  <span class="comment">//    b = [b0, b1] = [u0, v0, u1, v1],</span></div>
<div class="line"><a name="l00756"></a><span class="lineno">  756</span>&#160;  <span class="comment">// such that b0^2 = a0, b1^2 = a1.</span></div>
<div class="line"><a name="l00757"></a><span class="lineno">  757</span>&#160;  <span class="comment">//</span></div>
<div class="line"><a name="l00758"></a><span class="lineno">  758</span>&#160;  <span class="comment">// To derive the formula for the complex square roots, let&#39;s consider the equation for</span></div>
<div class="line"><a name="l00759"></a><span class="lineno">  759</span>&#160;  <span class="comment">// a single complex square root of the number x + i*y. We want to find real numbers</span></div>
<div class="line"><a name="l00760"></a><span class="lineno">  760</span>&#160;  <span class="comment">// u and v such that</span></div>
<div class="line"><a name="l00761"></a><span class="lineno">  761</span>&#160;  <span class="comment">//    (u + i*v)^2 = x + i*y  &lt;=&gt;</span></div>
<div class="line"><a name="l00762"></a><span class="lineno">  762</span>&#160;  <span class="comment">//    u^2 - v^2 + i*2*u*v = x + i*v.</span></div>
<div class="line"><a name="l00763"></a><span class="lineno">  763</span>&#160;  <span class="comment">// By equating the real and imaginary parts we get:</span></div>
<div class="line"><a name="l00764"></a><span class="lineno">  764</span>&#160;  <span class="comment">//    u^2 - v^2 = x</span></div>
<div class="line"><a name="l00765"></a><span class="lineno">  765</span>&#160;  <span class="comment">//    2*u*v = y.</span></div>
<div class="line"><a name="l00766"></a><span class="lineno">  766</span>&#160;  <span class="comment">//</span></div>
<div class="line"><a name="l00767"></a><span class="lineno">  767</span>&#160;  <span class="comment">// For x &gt;= 0, this has the numerically stable solution</span></div>
<div class="line"><a name="l00768"></a><span class="lineno">  768</span>&#160;  <span class="comment">//    u = sqrt(0.5 * (x + sqrt(x^2 + y^2)))</span></div>
<div class="line"><a name="l00769"></a><span class="lineno">  769</span>&#160;  <span class="comment">//    v = 0.5 * (y / u)</span></div>
<div class="line"><a name="l00770"></a><span class="lineno">  770</span>&#160;  <span class="comment">// and for x &lt; 0,</span></div>
<div class="line"><a name="l00771"></a><span class="lineno">  771</span>&#160;  <span class="comment">//    v = sign(y) * sqrt(0.5 * (-x + sqrt(x^2 + y^2)))</span></div>
<div class="line"><a name="l00772"></a><span class="lineno">  772</span>&#160;  <span class="comment">//    u = 0.5 * (y / v)</span></div>
<div class="line"><a name="l00773"></a><span class="lineno">  773</span>&#160;  <span class="comment">//</span></div>
<div class="line"><a name="l00774"></a><span class="lineno">  774</span>&#160;  <span class="comment">//  To avoid unnecessary over- and underflow, we compute sqrt(x^2 + y^2) as</span></div>
<div class="line"><a name="l00775"></a><span class="lineno">  775</span>&#160;  <span class="comment">//     l = max(|x|, |y|) * sqrt(1 + (min(|x|, |y|) / max(|x|, |y|))^2) ,</span></div>
<div class="line"><a name="l00776"></a><span class="lineno">  776</span>&#160; </div>
<div class="line"><a name="l00777"></a><span class="lineno">  777</span>&#160;  <span class="comment">// In the following, without lack of generality, we have annotated the code, assuming</span></div>
<div class="line"><a name="l00778"></a><span class="lineno">  778</span>&#160;  <span class="comment">// that the input is a packet of 2 complex numbers.</span></div>
<div class="line"><a name="l00779"></a><span class="lineno">  779</span>&#160;  <span class="comment">//</span></div>
<div class="line"><a name="l00780"></a><span class="lineno">  780</span>&#160;  <span class="comment">// Step 1. Compute l = [l0, l0, l1, l1], where</span></div>
<div class="line"><a name="l00781"></a><span class="lineno">  781</span>&#160;  <span class="comment">//    l0 = sqrt(x0^2 + y0^2),  l1 = sqrt(x1^2 + y1^2)</span></div>
<div class="line"><a name="l00782"></a><span class="lineno">  782</span>&#160;  <span class="comment">// To avoid over- and underflow, we use the stable formula for each hypotenuse</span></div>
<div class="line"><a name="l00783"></a><span class="lineno">  783</span>&#160;  <span class="comment">//    l0 = (min0 == 0 ? max0 : max0 * sqrt(1 + (min0/max0)**2)),</span></div>
<div class="line"><a name="l00784"></a><span class="lineno">  784</span>&#160;  <span class="comment">// where max0 = max(|x0|, |y0|), min0 = min(|x0|, |y0|), and similarly for l1.</span></div>
<div class="line"><a name="l00785"></a><span class="lineno">  785</span>&#160; </div>
<div class="line"><a name="l00786"></a><span class="lineno">  786</span>&#160;  RealPacket a_abs = pabs(a.v);           <span class="comment">// [|x0|, |y0|, |x1|, |y1|]</span></div>
<div class="line"><a name="l00787"></a><span class="lineno">  787</span>&#160;  RealPacket a_abs_flip = pcplxflip(Packet(a_abs)).v; <span class="comment">// [|y0|, |x0|, |y1|, |x1|]</span></div>
<div class="line"><a name="l00788"></a><span class="lineno">  788</span>&#160;  RealPacket a_max = pmax(a_abs, a_abs_flip);</div>
<div class="line"><a name="l00789"></a><span class="lineno">  789</span>&#160;  RealPacket a_min = pmin(a_abs, a_abs_flip);</div>
<div class="line"><a name="l00790"></a><span class="lineno">  790</span>&#160;  RealPacket a_min_zero_mask = pcmp_eq(a_min, pzero(a_min));</div>
<div class="line"><a name="l00791"></a><span class="lineno">  791</span>&#160;  RealPacket a_max_zero_mask = pcmp_eq(a_max, pzero(a_max));</div>
<div class="line"><a name="l00792"></a><span class="lineno">  792</span>&#160;  RealPacket r = pdiv(a_min, a_max);</div>
<div class="line"><a name="l00793"></a><span class="lineno">  793</span>&#160;  <span class="keyword">const</span> RealPacket cst_one  = pset1&lt;RealPacket&gt;(RealScalar(1));</div>
<div class="line"><a name="l00794"></a><span class="lineno">  794</span>&#160;  RealPacket l = pmul(a_max, psqrt(padd(cst_one, pmul(r, r))));  <span class="comment">// [l0, l0, l1, l1]</span></div>
<div class="line"><a name="l00795"></a><span class="lineno">  795</span>&#160;  <span class="comment">// Set l to a_max if a_min is zero.</span></div>
<div class="line"><a name="l00796"></a><span class="lineno">  796</span>&#160;  l = pselect(a_min_zero_mask, a_max, l);</div>
<div class="line"><a name="l00797"></a><span class="lineno">  797</span>&#160; </div>
<div class="line"><a name="l00798"></a><span class="lineno">  798</span>&#160;  <span class="comment">// Step 2. Compute [rho0, *, rho1, *], where</span></div>
<div class="line"><a name="l00799"></a><span class="lineno">  799</span>&#160;  <span class="comment">// rho0 = sqrt(0.5 * (l0 + |x0|)), rho1 =  sqrt(0.5 * (l1 + |x1|))</span></div>
<div class="line"><a name="l00800"></a><span class="lineno">  800</span>&#160;  <span class="comment">// We don&#39;t care about the imaginary parts computed here. They will be overwritten later.</span></div>
<div class="line"><a name="l00801"></a><span class="lineno">  801</span>&#160;  <span class="keyword">const</span> RealPacket cst_half = pset1&lt;RealPacket&gt;(RealScalar(0.5));</div>
<div class="line"><a name="l00802"></a><span class="lineno">  802</span>&#160;  Packet rho;</div>
<div class="line"><a name="l00803"></a><span class="lineno">  803</span>&#160;  rho.v = psqrt(pmul(cst_half, padd(a_abs, l)));</div>
<div class="line"><a name="l00804"></a><span class="lineno">  804</span>&#160; </div>
<div class="line"><a name="l00805"></a><span class="lineno">  805</span>&#160;  <span class="comment">// Step 3. Compute [rho0, eta0, rho1, eta1], where</span></div>
<div class="line"><a name="l00806"></a><span class="lineno">  806</span>&#160;  <span class="comment">// eta0 = (y0 / l0) / 2, and eta1 = (y1 / l1) / 2.</span></div>
<div class="line"><a name="l00807"></a><span class="lineno">  807</span>&#160;  <span class="comment">// set eta = 0 of input is 0 + i0.</span></div>
<div class="line"><a name="l00808"></a><span class="lineno">  808</span>&#160;  RealPacket eta = pandnot(pmul(cst_half, pdiv(a.v, pcplxflip(rho).v)), a_max_zero_mask);</div>
<div class="line"><a name="l00809"></a><span class="lineno">  809</span>&#160;  RealPacket real_mask = peven_mask(a.v);</div>
<div class="line"><a name="l00810"></a><span class="lineno">  810</span>&#160;  Packet positive_real_result;</div>
<div class="line"><a name="l00811"></a><span class="lineno">  811</span>&#160;  <span class="comment">// Compute result for inputs with positive real part.</span></div>
<div class="line"><a name="l00812"></a><span class="lineno">  812</span>&#160;  positive_real_result.v = pselect(real_mask, rho.v, eta);</div>
<div class="line"><a name="l00813"></a><span class="lineno">  813</span>&#160; </div>
<div class="line"><a name="l00814"></a><span class="lineno">  814</span>&#160;  <span class="comment">// Step 4. Compute solution for inputs with negative real part:</span></div>
<div class="line"><a name="l00815"></a><span class="lineno">  815</span>&#160;  <span class="comment">//         [|eta0|, sign(y0)*rho0, |eta1|, sign(y1)*rho1]</span></div>
<div class="line"><a name="l00816"></a><span class="lineno">  816</span>&#160;  <span class="keyword">const</span> RealScalar neg_zero = RealScalar(numext::bit_cast&lt;float&gt;(0x80000000u));</div>
<div class="line"><a name="l00817"></a><span class="lineno">  817</span>&#160;  <span class="keyword">const</span> RealPacket cst_imag_sign_mask = pset1&lt;Packet&gt;(Scalar(RealScalar(0.0), neg_zero)).v;</div>
<div class="line"><a name="l00818"></a><span class="lineno">  818</span>&#160;  RealPacket imag_signs = pand(a.v, cst_imag_sign_mask);</div>
<div class="line"><a name="l00819"></a><span class="lineno">  819</span>&#160;  Packet negative_real_result;</div>
<div class="line"><a name="l00820"></a><span class="lineno">  820</span>&#160;  <span class="comment">// Notice that rho is positive, so taking it&#39;s absolute value is a noop.</span></div>
<div class="line"><a name="l00821"></a><span class="lineno">  821</span>&#160;  negative_real_result.v = por(pabs(pcplxflip(positive_real_result).v), imag_signs);</div>
<div class="line"><a name="l00822"></a><span class="lineno">  822</span>&#160; </div>
<div class="line"><a name="l00823"></a><span class="lineno">  823</span>&#160;  <span class="comment">// Step 5. Select solution branch based on the sign of the real parts.</span></div>
<div class="line"><a name="l00824"></a><span class="lineno">  824</span>&#160;  Packet negative_real_mask;</div>
<div class="line"><a name="l00825"></a><span class="lineno">  825</span>&#160;  negative_real_mask.v = pcmp_lt(pand(real_mask, a.v), pzero(a.v));</div>
<div class="line"><a name="l00826"></a><span class="lineno">  826</span>&#160;  negative_real_mask.v = por(negative_real_mask.v, pcplxflip(negative_real_mask).v);</div>
<div class="line"><a name="l00827"></a><span class="lineno">  827</span>&#160;  Packet result = pselect(negative_real_mask, negative_real_result, positive_real_result);</div>
<div class="line"><a name="l00828"></a><span class="lineno">  828</span>&#160; </div>
<div class="line"><a name="l00829"></a><span class="lineno">  829</span>&#160;  <span class="comment">// Step 6. Handle special cases for infinities:</span></div>
<div class="line"><a name="l00830"></a><span class="lineno">  830</span>&#160;  <span class="comment">// * If z is (x,+∞), the result is (+∞,+∞) even if x is NaN</span></div>
<div class="line"><a name="l00831"></a><span class="lineno">  831</span>&#160;  <span class="comment">// * If z is (x,-∞), the result is (+∞,-∞) even if x is NaN</span></div>
<div class="line"><a name="l00832"></a><span class="lineno">  832</span>&#160;  <span class="comment">// * If z is (-∞,y), the result is (0*|y|,+∞) for finite or NaN y</span></div>
<div class="line"><a name="l00833"></a><span class="lineno">  833</span>&#160;  <span class="comment">// * If z is (+∞,y), the result is (+∞,0*|y|) for finite or NaN y</span></div>
<div class="line"><a name="l00834"></a><span class="lineno">  834</span>&#160;  <span class="keyword">const</span> RealPacket cst_pos_inf = pset1&lt;RealPacket&gt;(NumTraits&lt;RealScalar&gt;::infinity());</div>
<div class="line"><a name="l00835"></a><span class="lineno">  835</span>&#160;  Packet is_inf;</div>
<div class="line"><a name="l00836"></a><span class="lineno">  836</span>&#160;  is_inf.v = pcmp_eq(a_abs, cst_pos_inf);</div>
<div class="line"><a name="l00837"></a><span class="lineno">  837</span>&#160;  Packet is_real_inf;</div>
<div class="line"><a name="l00838"></a><span class="lineno">  838</span>&#160;  is_real_inf.v = pand(is_inf.v, real_mask);</div>
<div class="line"><a name="l00839"></a><span class="lineno">  839</span>&#160;  is_real_inf = por(is_real_inf, pcplxflip(is_real_inf));</div>
<div class="line"><a name="l00840"></a><span class="lineno">  840</span>&#160;  <span class="comment">// prepare packet of (+∞,0*|y|) or (0*|y|,+∞), depending on the sign of the infinite real part.</span></div>
<div class="line"><a name="l00841"></a><span class="lineno">  841</span>&#160;  Packet real_inf_result;</div>
<div class="line"><a name="l00842"></a><span class="lineno">  842</span>&#160;  real_inf_result.v = pmul(a_abs, pset1&lt;Packet&gt;(Scalar(RealScalar(1.0), RealScalar(0.0))).v);</div>
<div class="line"><a name="l00843"></a><span class="lineno">  843</span>&#160;  real_inf_result.v = pselect(negative_real_mask.v, pcplxflip(real_inf_result).v, real_inf_result.v);</div>
<div class="line"><a name="l00844"></a><span class="lineno">  844</span>&#160;  <span class="comment">// prepare packet of (+∞,+∞) or (+∞,-∞), depending on the sign of the infinite imaginary part.</span></div>
<div class="line"><a name="l00845"></a><span class="lineno">  845</span>&#160;  Packet is_imag_inf;</div>
<div class="line"><a name="l00846"></a><span class="lineno">  846</span>&#160;  is_imag_inf.v = pandnot(is_inf.v, real_mask);</div>
<div class="line"><a name="l00847"></a><span class="lineno">  847</span>&#160;  is_imag_inf = por(is_imag_inf, pcplxflip(is_imag_inf));</div>
<div class="line"><a name="l00848"></a><span class="lineno">  848</span>&#160;  Packet imag_inf_result;</div>
<div class="line"><a name="l00849"></a><span class="lineno">  849</span>&#160;  imag_inf_result.v = por(pand(cst_pos_inf, real_mask), pandnot(a.v, real_mask));</div>
<div class="line"><a name="l00850"></a><span class="lineno">  850</span>&#160; </div>
<div class="line"><a name="l00851"></a><span class="lineno">  851</span>&#160;  <span class="keywordflow">return</span>  pselect(is_imag_inf, imag_inf_result,</div>
<div class="line"><a name="l00852"></a><span class="lineno">  852</span>&#160;                  pselect(is_real_inf, real_inf_result,result));</div>
<div class="line"><a name="l00853"></a><span class="lineno">  853</span>&#160;}</div>
<div class="line"><a name="l00854"></a><span class="lineno">  854</span>&#160; </div>
<div class="line"><a name="l00855"></a><span class="lineno">  855</span>&#160;<span class="comment">// TODO(rmlarsen): The following set of utilities for double word arithmetic</span></div>
<div class="line"><a name="l00856"></a><span class="lineno">  856</span>&#160;<span class="comment">// should perhaps be refactored as a separate file, since it would be generally</span></div>
<div class="line"><a name="l00857"></a><span class="lineno">  857</span>&#160;<span class="comment">// useful for special function implementation etc. Writing the algorithms in</span></div>
<div class="line"><a name="l00858"></a><span class="lineno">  858</span>&#160;<span class="comment">// terms if a double word type would also make the code more readable.</span></div>
<div class="line"><a name="l00859"></a><span class="lineno">  859</span>&#160; </div>
<div class="line"><a name="l00860"></a><span class="lineno">  860</span>&#160;<span class="comment">// This function splits x into the nearest integer n and fractional part r,</span></div>
<div class="line"><a name="l00861"></a><span class="lineno">  861</span>&#160;<span class="comment">// such that x = n + r holds exactly.</span></div>
<div class="line"><a name="l00862"></a><span class="lineno">  862</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00863"></a><span class="lineno">  863</span>&#160;EIGEN_STRONG_INLINE</div>
<div class="line"><a name="l00864"></a><span class="lineno">  864</span>&#160;<span class="keywordtype">void</span> absolute_split(<span class="keyword">const</span> Packet&amp; x, Packet&amp; n, Packet&amp; r) {</div>
<div class="line"><a name="l00865"></a><span class="lineno">  865</span>&#160;  n = pround(x);</div>
<div class="line"><a name="l00866"></a><span class="lineno">  866</span>&#160;  r = psub(x, n);</div>
<div class="line"><a name="l00867"></a><span class="lineno">  867</span>&#160;}</div>
<div class="line"><a name="l00868"></a><span class="lineno">  868</span>&#160; </div>
<div class="line"><a name="l00869"></a><span class="lineno">  869</span>&#160;<span class="comment">// This function computes the sum {s, r}, such that x + y = s_hi + s_lo</span></div>
<div class="line"><a name="l00870"></a><span class="lineno">  870</span>&#160;<span class="comment">// holds exactly, and s_hi = fl(x+y), if |x| &gt;= |y|.</span></div>
<div class="line"><a name="l00871"></a><span class="lineno">  871</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00872"></a><span class="lineno">  872</span>&#160;EIGEN_STRONG_INLINE</div>
<div class="line"><a name="l00873"></a><span class="lineno">  873</span>&#160;<span class="keywordtype">void</span> fast_twosum(<span class="keyword">const</span> Packet&amp; x, <span class="keyword">const</span> Packet&amp; y, Packet&amp; s_hi, Packet&amp; s_lo) {</div>
<div class="line"><a name="l00874"></a><span class="lineno">  874</span>&#160;  s_hi = padd(x, y);</div>
<div class="line"><a name="l00875"></a><span class="lineno">  875</span>&#160;  <span class="keyword">const</span> Packet t = psub(s_hi, x);</div>
<div class="line"><a name="l00876"></a><span class="lineno">  876</span>&#160;  s_lo = psub(y, t);</div>
<div class="line"><a name="l00877"></a><span class="lineno">  877</span>&#160;}</div>
<div class="line"><a name="l00878"></a><span class="lineno">  878</span>&#160; </div>
<div class="line"><a name="l00879"></a><span class="lineno">  879</span>&#160;<span class="preprocessor">#ifdef EIGEN_HAS_SINGLE_INSTRUCTION_MADD</span></div>
<div class="line"><a name="l00880"></a><span class="lineno">  880</span>&#160;<span class="comment">// This function implements the extended precision product of</span></div>
<div class="line"><a name="l00881"></a><span class="lineno">  881</span>&#160;<span class="comment">// a pair of floating point numbers. Given {x, y}, it computes the pair</span></div>
<div class="line"><a name="l00882"></a><span class="lineno">  882</span>&#160;<span class="comment">// {p_hi, p_lo} such that x * y = p_hi + p_lo holds exactly and</span></div>
<div class="line"><a name="l00883"></a><span class="lineno">  883</span>&#160;<span class="comment">// p_hi = fl(x * y).</span></div>
<div class="line"><a name="l00884"></a><span class="lineno">  884</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00885"></a><span class="lineno">  885</span>&#160;EIGEN_STRONG_INLINE</div>
<div class="line"><a name="l00886"></a><span class="lineno">  886</span>&#160;<span class="keywordtype">void</span> twoprod(<span class="keyword">const</span> Packet&amp; x, <span class="keyword">const</span> Packet&amp; y,</div>
<div class="line"><a name="l00887"></a><span class="lineno">  887</span>&#160;             Packet&amp; p_hi, Packet&amp; p_lo) {</div>
<div class="line"><a name="l00888"></a><span class="lineno">  888</span>&#160;  p_hi = pmul(x, y);</div>
<div class="line"><a name="l00889"></a><span class="lineno">  889</span>&#160;  p_lo = pmadd(x, y, pnegate(p_hi));</div>
<div class="line"><a name="l00890"></a><span class="lineno">  890</span>&#160;}</div>
<div class="line"><a name="l00891"></a><span class="lineno">  891</span>&#160; </div>
<div class="line"><a name="l00892"></a><span class="lineno">  892</span>&#160;<span class="preprocessor">#else</span></div>
<div class="line"><a name="l00893"></a><span class="lineno">  893</span>&#160; </div>
<div class="line"><a name="l00894"></a><span class="lineno">  894</span>&#160;<span class="comment">// This function implements the Veltkamp splitting. Given a floating point</span></div>
<div class="line"><a name="l00895"></a><span class="lineno">  895</span>&#160;<span class="comment">// number x it returns the pair {x_hi, x_lo} such that x_hi + x_lo = x holds</span></div>
<div class="line"><a name="l00896"></a><span class="lineno">  896</span>&#160;<span class="comment">// exactly and that half of the significant of x fits in x_hi.</span></div>
<div class="line"><a name="l00897"></a><span class="lineno">  897</span>&#160;<span class="comment">// This is Algorithm 3 from Jean-Michel Muller, &quot;Elementary Functions&quot;,</span></div>
<div class="line"><a name="l00898"></a><span class="lineno">  898</span>&#160;<span class="comment">// 3rd edition, Birkh\&quot;auser, 2016.</span></div>
<div class="line"><a name="l00899"></a><span class="lineno">  899</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00900"></a><span class="lineno">  900</span>&#160;EIGEN_STRONG_INLINE</div>
<div class="line"><a name="l00901"></a><span class="lineno">  901</span>&#160;<span class="keywordtype">void</span> veltkamp_splitting(<span class="keyword">const</span> Packet&amp; x, Packet&amp; x_hi, Packet&amp; x_lo) {</div>
<div class="line"><a name="l00902"></a><span class="lineno">  902</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::type Scalar;</div>
<div class="line"><a name="l00903"></a><span class="lineno">  903</span>&#160;  EIGEN_CONSTEXPR <span class="keywordtype">int</span> shift = (NumTraits&lt;Scalar&gt;::digits() + 1) / 2;</div>
<div class="line"><a name="l00904"></a><span class="lineno">  904</span>&#160;  <span class="keyword">const</span> Scalar shift_scale = Scalar(uint64_t(1) &lt;&lt; shift);  <span class="comment">// Scalar constructor not necessarily constexpr.</span></div>
<div class="line"><a name="l00905"></a><span class="lineno">  905</span>&#160;  <span class="keyword">const</span> Packet gamma = pmul(pset1&lt;Packet&gt;(shift_scale + Scalar(1)), x);</div>
<div class="line"><a name="l00906"></a><span class="lineno">  906</span>&#160;  Packet rho = psub(x, gamma);</div>
<div class="line"><a name="l00907"></a><span class="lineno">  907</span>&#160;  x_hi = padd(rho, gamma);</div>
<div class="line"><a name="l00908"></a><span class="lineno">  908</span>&#160;  x_lo = psub(x, x_hi);</div>
<div class="line"><a name="l00909"></a><span class="lineno">  909</span>&#160;}</div>
<div class="line"><a name="l00910"></a><span class="lineno">  910</span>&#160; </div>
<div class="line"><a name="l00911"></a><span class="lineno">  911</span>&#160;<span class="comment">// This function implements Dekker&#39;s algorithm for products x * y.</span></div>
<div class="line"><a name="l00912"></a><span class="lineno">  912</span>&#160;<span class="comment">// Given floating point numbers {x, y} computes the pair</span></div>
<div class="line"><a name="l00913"></a><span class="lineno">  913</span>&#160;<span class="comment">// {p_hi, p_lo} such that x * y = p_hi + p_lo holds exactly and</span></div>
<div class="line"><a name="l00914"></a><span class="lineno">  914</span>&#160;<span class="comment">// p_hi = fl(x * y).</span></div>
<div class="line"><a name="l00915"></a><span class="lineno">  915</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00916"></a><span class="lineno">  916</span>&#160;EIGEN_STRONG_INLINE</div>
<div class="line"><a name="l00917"></a><span class="lineno">  917</span>&#160;<span class="keywordtype">void</span> twoprod(<span class="keyword">const</span> Packet&amp; x, <span class="keyword">const</span> Packet&amp; y,</div>
<div class="line"><a name="l00918"></a><span class="lineno">  918</span>&#160;             Packet&amp; p_hi, Packet&amp; p_lo) {</div>
<div class="line"><a name="l00919"></a><span class="lineno">  919</span>&#160;  Packet x_hi, x_lo, y_hi, y_lo;</div>
<div class="line"><a name="l00920"></a><span class="lineno">  920</span>&#160;  veltkamp_splitting(x, x_hi, x_lo);</div>
<div class="line"><a name="l00921"></a><span class="lineno">  921</span>&#160;  veltkamp_splitting(y, y_hi, y_lo);</div>
<div class="line"><a name="l00922"></a><span class="lineno">  922</span>&#160; </div>
<div class="line"><a name="l00923"></a><span class="lineno">  923</span>&#160;  p_hi = pmul(x, y);</div>
<div class="line"><a name="l00924"></a><span class="lineno">  924</span>&#160;  p_lo = pmadd(x_hi, y_hi, pnegate(p_hi));</div>
<div class="line"><a name="l00925"></a><span class="lineno">  925</span>&#160;  p_lo = pmadd(x_hi, y_lo, p_lo);</div>
<div class="line"><a name="l00926"></a><span class="lineno">  926</span>&#160;  p_lo = pmadd(x_lo, y_hi, p_lo);</div>
<div class="line"><a name="l00927"></a><span class="lineno">  927</span>&#160;  p_lo = pmadd(x_lo, y_lo, p_lo);</div>
<div class="line"><a name="l00928"></a><span class="lineno">  928</span>&#160;}</div>
<div class="line"><a name="l00929"></a><span class="lineno">  929</span>&#160; </div>
<div class="line"><a name="l00930"></a><span class="lineno">  930</span>&#160;<span class="preprocessor">#endif  </span><span class="comment">// EIGEN_HAS_SINGLE_INSTRUCTION_MADD</span></div>
<div class="line"><a name="l00931"></a><span class="lineno">  931</span>&#160; </div>
<div class="line"><a name="l00932"></a><span class="lineno">  932</span>&#160; </div>
<div class="line"><a name="l00933"></a><span class="lineno">  933</span>&#160;<span class="comment">// This function implements Dekker&#39;s algorithm for the addition</span></div>
<div class="line"><a name="l00934"></a><span class="lineno">  934</span>&#160;<span class="comment">// of two double word numbers represented by {x_hi, x_lo} and {y_hi, y_lo}.</span></div>
<div class="line"><a name="l00935"></a><span class="lineno">  935</span>&#160;<span class="comment">// It returns the result as a pair {s_hi, s_lo} such that</span></div>
<div class="line"><a name="l00936"></a><span class="lineno">  936</span>&#160;<span class="comment">// x_hi + x_lo + y_hi + y_lo = s_hi + s_lo holds exactly.</span></div>
<div class="line"><a name="l00937"></a><span class="lineno">  937</span>&#160;<span class="comment">// This is Algorithm 5 from Jean-Michel Muller, &quot;Elementary Functions&quot;,</span></div>
<div class="line"><a name="l00938"></a><span class="lineno">  938</span>&#160;<span class="comment">// 3rd edition, Birkh\&quot;auser, 2016.</span></div>
<div class="line"><a name="l00939"></a><span class="lineno">  939</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00940"></a><span class="lineno">  940</span>&#160;EIGEN_STRONG_INLINE</div>
<div class="line"><a name="l00941"></a><span class="lineno">  941</span>&#160;  <span class="keywordtype">void</span> twosum(<span class="keyword">const</span> Packet&amp; x_hi, <span class="keyword">const</span> Packet&amp; x_lo,</div>
<div class="line"><a name="l00942"></a><span class="lineno">  942</span>&#160;              <span class="keyword">const</span> Packet&amp; y_hi, <span class="keyword">const</span> Packet&amp; y_lo,</div>
<div class="line"><a name="l00943"></a><span class="lineno">  943</span>&#160;              Packet&amp; s_hi, Packet&amp; s_lo) {</div>
<div class="line"><a name="l00944"></a><span class="lineno">  944</span>&#160;  <span class="keyword">const</span> Packet x_greater_mask = pcmp_lt(pabs(y_hi), pabs(x_hi));</div>
<div class="line"><a name="l00945"></a><span class="lineno">  945</span>&#160;  Packet r_hi_1, r_lo_1;</div>
<div class="line"><a name="l00946"></a><span class="lineno">  946</span>&#160;  fast_twosum(x_hi, y_hi,r_hi_1, r_lo_1);</div>
<div class="line"><a name="l00947"></a><span class="lineno">  947</span>&#160;  Packet r_hi_2, r_lo_2;</div>
<div class="line"><a name="l00948"></a><span class="lineno">  948</span>&#160;  fast_twosum(y_hi, x_hi,r_hi_2, r_lo_2);</div>
<div class="line"><a name="l00949"></a><span class="lineno">  949</span>&#160;  <span class="keyword">const</span> Packet r_hi = pselect(x_greater_mask, r_hi_1, r_hi_2);</div>
<div class="line"><a name="l00950"></a><span class="lineno">  950</span>&#160; </div>
<div class="line"><a name="l00951"></a><span class="lineno">  951</span>&#160;  <span class="keyword">const</span> Packet s1 = padd(padd(y_lo, r_lo_1), x_lo);</div>
<div class="line"><a name="l00952"></a><span class="lineno">  952</span>&#160;  <span class="keyword">const</span> Packet s2 = padd(padd(x_lo, r_lo_2), y_lo);</div>
<div class="line"><a name="l00953"></a><span class="lineno">  953</span>&#160;  <span class="keyword">const</span> Packet s = pselect(x_greater_mask, s1, s2);</div>
<div class="line"><a name="l00954"></a><span class="lineno">  954</span>&#160; </div>
<div class="line"><a name="l00955"></a><span class="lineno">  955</span>&#160;  fast_twosum(r_hi, s, s_hi, s_lo);</div>
<div class="line"><a name="l00956"></a><span class="lineno">  956</span>&#160;}</div>
<div class="line"><a name="l00957"></a><span class="lineno">  957</span>&#160; </div>
<div class="line"><a name="l00958"></a><span class="lineno">  958</span>&#160;<span class="comment">// This is a version of twosum for double word numbers,</span></div>
<div class="line"><a name="l00959"></a><span class="lineno">  959</span>&#160;<span class="comment">// which assumes that |x_hi| &gt;= |y_hi|.</span></div>
<div class="line"><a name="l00960"></a><span class="lineno">  960</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00961"></a><span class="lineno">  961</span>&#160;EIGEN_STRONG_INLINE</div>
<div class="line"><a name="l00962"></a><span class="lineno">  962</span>&#160;  <span class="keywordtype">void</span> fast_twosum(<span class="keyword">const</span> Packet&amp; x_hi, <span class="keyword">const</span> Packet&amp; x_lo,</div>
<div class="line"><a name="l00963"></a><span class="lineno">  963</span>&#160;              <span class="keyword">const</span> Packet&amp; y_hi, <span class="keyword">const</span> Packet&amp; y_lo,</div>
<div class="line"><a name="l00964"></a><span class="lineno">  964</span>&#160;              Packet&amp; s_hi, Packet&amp; s_lo) {</div>
<div class="line"><a name="l00965"></a><span class="lineno">  965</span>&#160;  Packet r_hi, r_lo;</div>
<div class="line"><a name="l00966"></a><span class="lineno">  966</span>&#160;  fast_twosum(x_hi, y_hi, r_hi, r_lo);</div>
<div class="line"><a name="l00967"></a><span class="lineno">  967</span>&#160;  <span class="keyword">const</span> Packet s = padd(padd(y_lo, r_lo), x_lo);</div>
<div class="line"><a name="l00968"></a><span class="lineno">  968</span>&#160;  fast_twosum(r_hi, s, s_hi, s_lo);</div>
<div class="line"><a name="l00969"></a><span class="lineno">  969</span>&#160;}</div>
<div class="line"><a name="l00970"></a><span class="lineno">  970</span>&#160; </div>
<div class="line"><a name="l00971"></a><span class="lineno">  971</span>&#160;<span class="comment">// This is a version of twosum for adding a floating point number x to</span></div>
<div class="line"><a name="l00972"></a><span class="lineno">  972</span>&#160;<span class="comment">// double word number {y_hi, y_lo} number, with the assumption</span></div>
<div class="line"><a name="l00973"></a><span class="lineno">  973</span>&#160;<span class="comment">// that |x| &gt;= |y_hi|.</span></div>
<div class="line"><a name="l00974"></a><span class="lineno">  974</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00975"></a><span class="lineno">  975</span>&#160;EIGEN_STRONG_INLINE</div>
<div class="line"><a name="l00976"></a><span class="lineno">  976</span>&#160;<span class="keywordtype">void</span> fast_twosum(<span class="keyword">const</span> Packet&amp; x,</div>
<div class="line"><a name="l00977"></a><span class="lineno">  977</span>&#160;                 <span class="keyword">const</span> Packet&amp; y_hi, <span class="keyword">const</span> Packet&amp; y_lo,</div>
<div class="line"><a name="l00978"></a><span class="lineno">  978</span>&#160;                 Packet&amp; s_hi, Packet&amp; s_lo) {</div>
<div class="line"><a name="l00979"></a><span class="lineno">  979</span>&#160;  Packet r_hi, r_lo;</div>
<div class="line"><a name="l00980"></a><span class="lineno">  980</span>&#160;  fast_twosum(x, y_hi, r_hi, r_lo);</div>
<div class="line"><a name="l00981"></a><span class="lineno">  981</span>&#160;  <span class="keyword">const</span> Packet s = padd(y_lo, r_lo);</div>
<div class="line"><a name="l00982"></a><span class="lineno">  982</span>&#160;  fast_twosum(r_hi, s, s_hi, s_lo);</div>
<div class="line"><a name="l00983"></a><span class="lineno">  983</span>&#160;}</div>
<div class="line"><a name="l00984"></a><span class="lineno">  984</span>&#160; </div>
<div class="line"><a name="l00985"></a><span class="lineno">  985</span>&#160;<span class="comment">// This function implements the multiplication of a double word</span></div>
<div class="line"><a name="l00986"></a><span class="lineno">  986</span>&#160;<span class="comment">// number represented by {x_hi, x_lo} by a floating point number y.</span></div>
<div class="line"><a name="l00987"></a><span class="lineno">  987</span>&#160;<span class="comment">// It returns the result as a pair {p_hi, p_lo} such that</span></div>
<div class="line"><a name="l00988"></a><span class="lineno">  988</span>&#160;<span class="comment">// (x_hi + x_lo) * y = p_hi + p_lo hold with a relative error</span></div>
<div class="line"><a name="l00989"></a><span class="lineno">  989</span>&#160;<span class="comment">// of less than 2*2^{-2p}, where p is the number of significand bit</span></div>
<div class="line"><a name="l00990"></a><span class="lineno">  990</span>&#160;<span class="comment">// in the floating point type.</span></div>
<div class="line"><a name="l00991"></a><span class="lineno">  991</span>&#160;<span class="comment">// This is Algorithm 7 from Jean-Michel Muller, &quot;Elementary Functions&quot;,</span></div>
<div class="line"><a name="l00992"></a><span class="lineno">  992</span>&#160;<span class="comment">// 3rd edition, Birkh\&quot;auser, 2016.</span></div>
<div class="line"><a name="l00993"></a><span class="lineno">  993</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l00994"></a><span class="lineno">  994</span>&#160;EIGEN_STRONG_INLINE</div>
<div class="line"><a name="l00995"></a><span class="lineno">  995</span>&#160;<span class="keywordtype">void</span> twoprod(<span class="keyword">const</span> Packet&amp; x_hi, <span class="keyword">const</span> Packet&amp; x_lo, <span class="keyword">const</span> Packet&amp; y,</div>
<div class="line"><a name="l00996"></a><span class="lineno">  996</span>&#160;             Packet&amp; p_hi, Packet&amp; p_lo) {</div>
<div class="line"><a name="l00997"></a><span class="lineno">  997</span>&#160;  Packet c_hi, c_lo1;</div>
<div class="line"><a name="l00998"></a><span class="lineno">  998</span>&#160;  twoprod(x_hi, y, c_hi, c_lo1);</div>
<div class="line"><a name="l00999"></a><span class="lineno">  999</span>&#160;  <span class="keyword">const</span> Packet c_lo2 = pmul(x_lo, y);</div>
<div class="line"><a name="l01000"></a><span class="lineno"> 1000</span>&#160;  Packet t_hi, t_lo1;</div>
<div class="line"><a name="l01001"></a><span class="lineno"> 1001</span>&#160;  fast_twosum(c_hi, c_lo2, t_hi, t_lo1);</div>
<div class="line"><a name="l01002"></a><span class="lineno"> 1002</span>&#160;  <span class="keyword">const</span> Packet t_lo2 = padd(t_lo1, c_lo1);</div>
<div class="line"><a name="l01003"></a><span class="lineno"> 1003</span>&#160;  fast_twosum(t_hi, t_lo2, p_hi, p_lo);</div>
<div class="line"><a name="l01004"></a><span class="lineno"> 1004</span>&#160;}</div>
<div class="line"><a name="l01005"></a><span class="lineno"> 1005</span>&#160; </div>
<div class="line"><a name="l01006"></a><span class="lineno"> 1006</span>&#160;<span class="comment">// This function implements the multiplication of two double word</span></div>
<div class="line"><a name="l01007"></a><span class="lineno"> 1007</span>&#160;<span class="comment">// numbers represented by {x_hi, x_lo} and {y_hi, y_lo}.</span></div>
<div class="line"><a name="l01008"></a><span class="lineno"> 1008</span>&#160;<span class="comment">// It returns the result as a pair {p_hi, p_lo} such that</span></div>
<div class="line"><a name="l01009"></a><span class="lineno"> 1009</span>&#160;<span class="comment">// (x_hi + x_lo) * (y_hi + y_lo) = p_hi + p_lo holds with a relative error</span></div>
<div class="line"><a name="l01010"></a><span class="lineno"> 1010</span>&#160;<span class="comment">// of less than 2*2^{-2p}, where p is the number of significand bit</span></div>
<div class="line"><a name="l01011"></a><span class="lineno"> 1011</span>&#160;<span class="comment">// in the floating point type.</span></div>
<div class="line"><a name="l01012"></a><span class="lineno"> 1012</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l01013"></a><span class="lineno"> 1013</span>&#160;EIGEN_STRONG_INLINE</div>
<div class="line"><a name="l01014"></a><span class="lineno"> 1014</span>&#160;<span class="keywordtype">void</span> twoprod(<span class="keyword">const</span> Packet&amp; x_hi, <span class="keyword">const</span> Packet&amp; x_lo,</div>
<div class="line"><a name="l01015"></a><span class="lineno"> 1015</span>&#160;             <span class="keyword">const</span> Packet&amp; y_hi, <span class="keyword">const</span> Packet&amp; y_lo,</div>
<div class="line"><a name="l01016"></a><span class="lineno"> 1016</span>&#160;             Packet&amp; p_hi, Packet&amp; p_lo) {</div>
<div class="line"><a name="l01017"></a><span class="lineno"> 1017</span>&#160;  Packet p_hi_hi, p_hi_lo;</div>
<div class="line"><a name="l01018"></a><span class="lineno"> 1018</span>&#160;  twoprod(x_hi, x_lo, y_hi, p_hi_hi, p_hi_lo);</div>
<div class="line"><a name="l01019"></a><span class="lineno"> 1019</span>&#160;  Packet p_lo_hi, p_lo_lo;</div>
<div class="line"><a name="l01020"></a><span class="lineno"> 1020</span>&#160;  twoprod(x_hi, x_lo, y_lo, p_lo_hi, p_lo_lo);</div>
<div class="line"><a name="l01021"></a><span class="lineno"> 1021</span>&#160;  fast_twosum(p_hi_hi, p_hi_lo, p_lo_hi, p_lo_lo, p_hi, p_lo);</div>
<div class="line"><a name="l01022"></a><span class="lineno"> 1022</span>&#160;}</div>
<div class="line"><a name="l01023"></a><span class="lineno"> 1023</span>&#160; </div>
<div class="line"><a name="l01024"></a><span class="lineno"> 1024</span>&#160;<span class="comment">// This function computes the reciprocal of a floating point number</span></div>
<div class="line"><a name="l01025"></a><span class="lineno"> 1025</span>&#160;<span class="comment">// with extra precision and returns the result as a double word.</span></div>
<div class="line"><a name="l01026"></a><span class="lineno"> 1026</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l01027"></a><span class="lineno"> 1027</span>&#160;<span class="keywordtype">void</span> doubleword_reciprocal(<span class="keyword">const</span> Packet&amp; x, Packet&amp; recip_hi, Packet&amp; recip_lo) {</div>
<div class="line"><a name="l01028"></a><span class="lineno"> 1028</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::type Scalar;</div>
<div class="line"><a name="l01029"></a><span class="lineno"> 1029</span>&#160;  <span class="comment">// 1. Approximate the reciprocal as the reciprocal of the high order element.</span></div>
<div class="line"><a name="l01030"></a><span class="lineno"> 1030</span>&#160;  Packet approx_recip = prsqrt(x);</div>
<div class="line"><a name="l01031"></a><span class="lineno"> 1031</span>&#160;  approx_recip = pmul(approx_recip, approx_recip);</div>
<div class="line"><a name="l01032"></a><span class="lineno"> 1032</span>&#160; </div>
<div class="line"><a name="l01033"></a><span class="lineno"> 1033</span>&#160;  <span class="comment">// 2. Run one step of Newton-Raphson iteration in double word arithmetic</span></div>
<div class="line"><a name="l01034"></a><span class="lineno"> 1034</span>&#160;  <span class="comment">// to get the bottom half. The NR iteration for reciprocal of &#39;a&#39; is</span></div>
<div class="line"><a name="l01035"></a><span class="lineno"> 1035</span>&#160;  <span class="comment">//    x_{i+1} = x_i * (2 - a * x_i)</span></div>
<div class="line"><a name="l01036"></a><span class="lineno"> 1036</span>&#160; </div>
<div class="line"><a name="l01037"></a><span class="lineno"> 1037</span>&#160;  <span class="comment">// -a*x_i</span></div>
<div class="line"><a name="l01038"></a><span class="lineno"> 1038</span>&#160;  Packet t1_hi, t1_lo;</div>
<div class="line"><a name="l01039"></a><span class="lineno"> 1039</span>&#160;  twoprod(pnegate(x), approx_recip, t1_hi, t1_lo);</div>
<div class="line"><a name="l01040"></a><span class="lineno"> 1040</span>&#160;  <span class="comment">// 2 - a*x_i</span></div>
<div class="line"><a name="l01041"></a><span class="lineno"> 1041</span>&#160;  Packet t2_hi, t2_lo;</div>
<div class="line"><a name="l01042"></a><span class="lineno"> 1042</span>&#160;  fast_twosum(pset1&lt;Packet&gt;(Scalar(2)), t1_hi, t2_hi, t2_lo);</div>
<div class="line"><a name="l01043"></a><span class="lineno"> 1043</span>&#160;  Packet t3_hi, t3_lo;</div>
<div class="line"><a name="l01044"></a><span class="lineno"> 1044</span>&#160;  fast_twosum(t2_hi, padd(t2_lo, t1_lo), t3_hi, t3_lo);</div>
<div class="line"><a name="l01045"></a><span class="lineno"> 1045</span>&#160;  <span class="comment">// x_i * (2 - a * x_i)</span></div>
<div class="line"><a name="l01046"></a><span class="lineno"> 1046</span>&#160;  twoprod(t3_hi, t3_lo, approx_recip, recip_hi, recip_lo);</div>
<div class="line"><a name="l01047"></a><span class="lineno"> 1047</span>&#160;}</div>
<div class="line"><a name="l01048"></a><span class="lineno"> 1048</span>&#160; </div>
<div class="line"><a name="l01049"></a><span class="lineno"> 1049</span>&#160; </div>
<div class="line"><a name="l01050"></a><span class="lineno"> 1050</span>&#160;<span class="comment">// This function computes log2(x) and returns the result as a double word.</span></div>
<div class="line"><a name="l01051"></a><span class="lineno"> 1051</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> Scalar&gt;</div>
<div class="line"><a name="l01052"></a><span class="lineno"> 1052</span>&#160;<span class="keyword">struct </span>accurate_log2 {</div>
<div class="line"><a name="l01053"></a><span class="lineno"> 1053</span>&#160;  <span class="keyword">template</span> &lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l01054"></a><span class="lineno"> 1054</span>&#160;  EIGEN_STRONG_INLINE</div>
<div class="line"><a name="l01055"></a><span class="lineno"> 1055</span>&#160;  <span class="keywordtype">void</span> operator()(<span class="keyword">const</span> Packet&amp; x, Packet&amp; log2_x_hi, Packet&amp; log2_x_lo) {</div>
<div class="line"><a name="l01056"></a><span class="lineno"> 1056</span>&#160;    log2_x_hi = plog2(x);</div>
<div class="line"><a name="l01057"></a><span class="lineno"> 1057</span>&#160;    log2_x_lo = pzero(x);</div>
<div class="line"><a name="l01058"></a><span class="lineno"> 1058</span>&#160;  }</div>
<div class="line"><a name="l01059"></a><span class="lineno"> 1059</span>&#160;};</div>
<div class="line"><a name="l01060"></a><span class="lineno"> 1060</span>&#160; </div>
<div class="line"><a name="l01061"></a><span class="lineno"> 1061</span>&#160;<span class="comment">// This specialization uses a more accurate algorithm to compute log2(x) for</span></div>
<div class="line"><a name="l01062"></a><span class="lineno"> 1062</span>&#160;<span class="comment">// floats in [1/sqrt(2);sqrt(2)] with a relative accuracy of ~6.42e-10.</span></div>
<div class="line"><a name="l01063"></a><span class="lineno"> 1063</span>&#160;<span class="comment">// This additional accuracy is needed to counter the error-magnification</span></div>
<div class="line"><a name="l01064"></a><span class="lineno"> 1064</span>&#160;<span class="comment">// inherent in multiplying by a potentially large exponent in pow(x,y).</span></div>
<div class="line"><a name="l01065"></a><span class="lineno"> 1065</span>&#160;<span class="comment">// The minimax polynomial used was calculated using the Sollya tool.</span></div>
<div class="line"><a name="l01066"></a><span class="lineno"> 1066</span>&#160;<span class="comment">// See sollya.org.</span></div>
<div class="line"><a name="l01067"></a><span class="lineno"> 1067</span>&#160;<span class="keyword">template</span> &lt;&gt;</div>
<div class="line"><a name="l01068"></a><span class="lineno"> 1068</span>&#160;<span class="keyword">struct </span>accurate_log2&lt;float&gt; {</div>
<div class="line"><a name="l01069"></a><span class="lineno"> 1069</span>&#160;  <span class="keyword">template</span> &lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l01070"></a><span class="lineno"> 1070</span>&#160;  EIGEN_STRONG_INLINE</div>
<div class="line"><a name="l01071"></a><span class="lineno"> 1071</span>&#160;  <span class="keywordtype">void</span> operator()(<span class="keyword">const</span> Packet&amp; z, Packet&amp; log2_x_hi, Packet&amp; log2_x_lo) {</div>
<div class="line"><a name="l01072"></a><span class="lineno"> 1072</span>&#160;    <span class="comment">// The function log(1+x)/x is approximated in the interval</span></div>
<div class="line"><a name="l01073"></a><span class="lineno"> 1073</span>&#160;    <span class="comment">// [1/sqrt(2)-1;sqrt(2)-1] by a degree 10 polynomial of the form</span></div>
<div class="line"><a name="l01074"></a><span class="lineno"> 1074</span>&#160;    <span class="comment">//  Q(x) = (C0 + x * (C1 + x * (C2 + x * (C3 + x * P(x))))),</span></div>
<div class="line"><a name="l01075"></a><span class="lineno"> 1075</span>&#160;    <span class="comment">// where the degree 6 polynomial P(x) is evaluated in single precision,</span></div>
<div class="line"><a name="l01076"></a><span class="lineno"> 1076</span>&#160;    <span class="comment">// while the remaining 4 terms of Q(x), as well as the final multiplication by x</span></div>
<div class="line"><a name="l01077"></a><span class="lineno"> 1077</span>&#160;    <span class="comment">// to reconstruct log(1+x) are evaluated in extra precision using</span></div>
<div class="line"><a name="l01078"></a><span class="lineno"> 1078</span>&#160;    <span class="comment">// double word arithmetic. C0 through C3 are extra precise constants</span></div>
<div class="line"><a name="l01079"></a><span class="lineno"> 1079</span>&#160;    <span class="comment">// stored as double words.</span></div>
<div class="line"><a name="l01080"></a><span class="lineno"> 1080</span>&#160;    <span class="comment">//</span></div>
<div class="line"><a name="l01081"></a><span class="lineno"> 1081</span>&#160;    <span class="comment">// The polynomial coefficients were calculated using Sollya commands:</span></div>
<div class="line"><a name="l01082"></a><span class="lineno"> 1082</span>&#160;    <span class="comment">// &gt; n = 10;</span></div>
<div class="line"><a name="l01083"></a><span class="lineno"> 1083</span>&#160;    <span class="comment">// &gt; f = log2(1+x)/x;</span></div>
<div class="line"><a name="l01084"></a><span class="lineno"> 1084</span>&#160;    <span class="comment">// &gt; interval = [sqrt(0.5)-1;sqrt(2)-1];</span></div>
<div class="line"><a name="l01085"></a><span class="lineno"> 1085</span>&#160;    <span class="comment">// &gt; p = fpminimax(f,n,[|double,double,double,double,single...|],interval,relative,floating);</span></div>
<div class="line"><a name="l01086"></a><span class="lineno"> 1086</span>&#160;    </div>
<div class="line"><a name="l01087"></a><span class="lineno"> 1087</span>&#160;    <span class="keyword">const</span> Packet p6 = pset1&lt;Packet&gt;( 9.703654795885e-2f);</div>
<div class="line"><a name="l01088"></a><span class="lineno"> 1088</span>&#160;    <span class="keyword">const</span> Packet p5 = pset1&lt;Packet&gt;(-0.1690667718648f);</div>
<div class="line"><a name="l01089"></a><span class="lineno"> 1089</span>&#160;    <span class="keyword">const</span> Packet p4 = pset1&lt;Packet&gt;( 0.1720575392246f);</div>
<div class="line"><a name="l01090"></a><span class="lineno"> 1090</span>&#160;    <span class="keyword">const</span> Packet p3 = pset1&lt;Packet&gt;(-0.1789081543684f);</div>
<div class="line"><a name="l01091"></a><span class="lineno"> 1091</span>&#160;    <span class="keyword">const</span> Packet p2 = pset1&lt;Packet&gt;( 0.2050433009862f);</div>
<div class="line"><a name="l01092"></a><span class="lineno"> 1092</span>&#160;    <span class="keyword">const</span> Packet p1 = pset1&lt;Packet&gt;(-0.2404672354459f);</div>
<div class="line"><a name="l01093"></a><span class="lineno"> 1093</span>&#160;    <span class="keyword">const</span> Packet p0 = pset1&lt;Packet&gt;( 0.2885761857032f);</div>
<div class="line"><a name="l01094"></a><span class="lineno"> 1094</span>&#160; </div>
<div class="line"><a name="l01095"></a><span class="lineno"> 1095</span>&#160;    <span class="keyword">const</span> Packet C3_hi = pset1&lt;Packet&gt;(-0.360674142838f);</div>
<div class="line"><a name="l01096"></a><span class="lineno"> 1096</span>&#160;    <span class="keyword">const</span> Packet C3_lo = pset1&lt;Packet&gt;(-6.13283912543e-09f);</div>
<div class="line"><a name="l01097"></a><span class="lineno"> 1097</span>&#160;    <span class="keyword">const</span> Packet C2_hi = pset1&lt;Packet&gt;(0.480897903442f);</div>
<div class="line"><a name="l01098"></a><span class="lineno"> 1098</span>&#160;    <span class="keyword">const</span> Packet C2_lo = pset1&lt;Packet&gt;(-1.44861207474e-08f);</div>
<div class="line"><a name="l01099"></a><span class="lineno"> 1099</span>&#160;    <span class="keyword">const</span> Packet C1_hi = pset1&lt;Packet&gt;(-0.721347510815f);</div>
<div class="line"><a name="l01100"></a><span class="lineno"> 1100</span>&#160;    <span class="keyword">const</span> Packet C1_lo = pset1&lt;Packet&gt;(-4.84483164698e-09f);</div>
<div class="line"><a name="l01101"></a><span class="lineno"> 1101</span>&#160;    <span class="keyword">const</span> Packet C0_hi = pset1&lt;Packet&gt;(1.44269502163f);</div>
<div class="line"><a name="l01102"></a><span class="lineno"> 1102</span>&#160;    <span class="keyword">const</span> Packet C0_lo = pset1&lt;Packet&gt;(2.01711713999e-08f);</div>
<div class="line"><a name="l01103"></a><span class="lineno"> 1103</span>&#160;    <span class="keyword">const</span> Packet one = pset1&lt;Packet&gt;(1.0f);</div>
<div class="line"><a name="l01104"></a><span class="lineno"> 1104</span>&#160; </div>
<div class="line"><a name="l01105"></a><span class="lineno"> 1105</span>&#160;    <span class="keyword">const</span> Packet x = psub(z, one);</div>
<div class="line"><a name="l01106"></a><span class="lineno"> 1106</span>&#160;    <span class="comment">// Evaluate P(x) in working precision.</span></div>
<div class="line"><a name="l01107"></a><span class="lineno"> 1107</span>&#160;    <span class="comment">// We evaluate it in multiple parts to improve instruction level</span></div>
<div class="line"><a name="l01108"></a><span class="lineno"> 1108</span>&#160;    <span class="comment">// parallelism.</span></div>
<div class="line"><a name="l01109"></a><span class="lineno"> 1109</span>&#160;    Packet x2 = pmul(x,x);</div>
<div class="line"><a name="l01110"></a><span class="lineno"> 1110</span>&#160;    Packet p_even = pmadd(p6, x2, p4);</div>
<div class="line"><a name="l01111"></a><span class="lineno"> 1111</span>&#160;    p_even = pmadd(p_even, x2, p2);</div>
<div class="line"><a name="l01112"></a><span class="lineno"> 1112</span>&#160;    p_even = pmadd(p_even, x2, p0);</div>
<div class="line"><a name="l01113"></a><span class="lineno"> 1113</span>&#160;    Packet p_odd = pmadd(p5, x2, p3);</div>
<div class="line"><a name="l01114"></a><span class="lineno"> 1114</span>&#160;    p_odd = pmadd(p_odd, x2, p1);</div>
<div class="line"><a name="l01115"></a><span class="lineno"> 1115</span>&#160;    Packet p = pmadd(p_odd, x, p_even);</div>
<div class="line"><a name="l01116"></a><span class="lineno"> 1116</span>&#160; </div>
<div class="line"><a name="l01117"></a><span class="lineno"> 1117</span>&#160;    <span class="comment">// Now evaluate the low-order tems of Q(x) in double word precision.</span></div>
<div class="line"><a name="l01118"></a><span class="lineno"> 1118</span>&#160;    <span class="comment">// In the following, due to the alternating signs and the fact that</span></div>
<div class="line"><a name="l01119"></a><span class="lineno"> 1119</span>&#160;    <span class="comment">// |x| &lt; sqrt(2)-1, we can assume that |C*_hi| &gt;= q_i, and use</span></div>
<div class="line"><a name="l01120"></a><span class="lineno"> 1120</span>&#160;    <span class="comment">// fast_twosum instead of the slower twosum.</span></div>
<div class="line"><a name="l01121"></a><span class="lineno"> 1121</span>&#160;    Packet q_hi, q_lo;</div>
<div class="line"><a name="l01122"></a><span class="lineno"> 1122</span>&#160;    Packet t_hi, t_lo;</div>
<div class="line"><a name="l01123"></a><span class="lineno"> 1123</span>&#160;    <span class="comment">// C3 + x * p(x)</span></div>
<div class="line"><a name="l01124"></a><span class="lineno"> 1124</span>&#160;    twoprod(p, x, t_hi, t_lo);</div>
<div class="line"><a name="l01125"></a><span class="lineno"> 1125</span>&#160;    fast_twosum(C3_hi, C3_lo, t_hi, t_lo, q_hi, q_lo);</div>
<div class="line"><a name="l01126"></a><span class="lineno"> 1126</span>&#160;    <span class="comment">// C2 + x * p(x)</span></div>
<div class="line"><a name="l01127"></a><span class="lineno"> 1127</span>&#160;    twoprod(q_hi, q_lo, x, t_hi, t_lo);</div>
<div class="line"><a name="l01128"></a><span class="lineno"> 1128</span>&#160;    fast_twosum(C2_hi, C2_lo, t_hi, t_lo, q_hi, q_lo);</div>
<div class="line"><a name="l01129"></a><span class="lineno"> 1129</span>&#160;    <span class="comment">// C1 + x * p(x)</span></div>
<div class="line"><a name="l01130"></a><span class="lineno"> 1130</span>&#160;    twoprod(q_hi, q_lo, x, t_hi, t_lo);</div>
<div class="line"><a name="l01131"></a><span class="lineno"> 1131</span>&#160;    fast_twosum(C1_hi, C1_lo, t_hi, t_lo, q_hi, q_lo);</div>
<div class="line"><a name="l01132"></a><span class="lineno"> 1132</span>&#160;    <span class="comment">// C0 + x * p(x)</span></div>
<div class="line"><a name="l01133"></a><span class="lineno"> 1133</span>&#160;    twoprod(q_hi, q_lo, x, t_hi, t_lo);</div>
<div class="line"><a name="l01134"></a><span class="lineno"> 1134</span>&#160;    fast_twosum(C0_hi, C0_lo, t_hi, t_lo, q_hi, q_lo);</div>
<div class="line"><a name="l01135"></a><span class="lineno"> 1135</span>&#160; </div>
<div class="line"><a name="l01136"></a><span class="lineno"> 1136</span>&#160;    <span class="comment">// log(z) ~= x * Q(x)</span></div>
<div class="line"><a name="l01137"></a><span class="lineno"> 1137</span>&#160;    twoprod(q_hi, q_lo, x, log2_x_hi, log2_x_lo);</div>
<div class="line"><a name="l01138"></a><span class="lineno"> 1138</span>&#160;  }</div>
<div class="line"><a name="l01139"></a><span class="lineno"> 1139</span>&#160;};</div>
<div class="line"><a name="l01140"></a><span class="lineno"> 1140</span>&#160; </div>
<div class="line"><a name="l01141"></a><span class="lineno"> 1141</span>&#160;<span class="comment">// This specialization uses a more accurate algorithm to compute log2(x) for</span></div>
<div class="line"><a name="l01142"></a><span class="lineno"> 1142</span>&#160;<span class="comment">// floats in [1/sqrt(2);sqrt(2)] with a relative accuracy of ~1.27e-18.</span></div>
<div class="line"><a name="l01143"></a><span class="lineno"> 1143</span>&#160;<span class="comment">// This additional accuracy is needed to counter the error-magnification</span></div>
<div class="line"><a name="l01144"></a><span class="lineno"> 1144</span>&#160;<span class="comment">// inherent in multiplying by a potentially large exponent in pow(x,y).</span></div>
<div class="line"><a name="l01145"></a><span class="lineno"> 1145</span>&#160;<span class="comment">// The minimax polynomial used was calculated using the Sollya tool.</span></div>
<div class="line"><a name="l01146"></a><span class="lineno"> 1146</span>&#160;<span class="comment">// See sollya.org.</span></div>
<div class="line"><a name="l01147"></a><span class="lineno"> 1147</span>&#160; </div>
<div class="line"><a name="l01148"></a><span class="lineno"> 1148</span>&#160;<span class="keyword">template</span> &lt;&gt;</div>
<div class="line"><a name="l01149"></a><span class="lineno"> 1149</span>&#160;<span class="keyword">struct </span>accurate_log2&lt;double&gt; {</div>
<div class="line"><a name="l01150"></a><span class="lineno"> 1150</span>&#160;  <span class="keyword">template</span> &lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l01151"></a><span class="lineno"> 1151</span>&#160;  EIGEN_STRONG_INLINE</div>
<div class="line"><a name="l01152"></a><span class="lineno"> 1152</span>&#160;  <span class="keywordtype">void</span> operator()(<span class="keyword">const</span> Packet&amp; x, Packet&amp; log2_x_hi, Packet&amp; log2_x_lo) {</div>
<div class="line"><a name="l01153"></a><span class="lineno"> 1153</span>&#160;    <span class="comment">// We use a transformation of variables:</span></div>
<div class="line"><a name="l01154"></a><span class="lineno"> 1154</span>&#160;    <span class="comment">//    r = c * (x-1) / (x+1),</span></div>
<div class="line"><a name="l01155"></a><span class="lineno"> 1155</span>&#160;    <span class="comment">// such that</span></div>
<div class="line"><a name="l01156"></a><span class="lineno"> 1156</span>&#160;    <span class="comment">//    log2(x) = log2((1 + r/c) / (1 - r/c)) = f(r).</span></div>
<div class="line"><a name="l01157"></a><span class="lineno"> 1157</span>&#160;    <span class="comment">// The function f(r) can be approximated well using an odd polynomial</span></div>
<div class="line"><a name="l01158"></a><span class="lineno"> 1158</span>&#160;    <span class="comment">// of the form</span></div>
<div class="line"><a name="l01159"></a><span class="lineno"> 1159</span>&#160;    <span class="comment">//   P(r) = ((Q(r^2) * r^2 + C) * r^2 + 1) * r,</span></div>
<div class="line"><a name="l01160"></a><span class="lineno"> 1160</span>&#160;    <span class="comment">// For the implementation of log2&lt;double&gt; here, Q is of degree 6 with</span></div>
<div class="line"><a name="l01161"></a><span class="lineno"> 1161</span>&#160;    <span class="comment">// coefficient represented in working precision (double), while C is a</span></div>
<div class="line"><a name="l01162"></a><span class="lineno"> 1162</span>&#160;    <span class="comment">// constant represented in extra precision as a double word to achieve</span></div>
<div class="line"><a name="l01163"></a><span class="lineno"> 1163</span>&#160;    <span class="comment">// full accuracy.</span></div>
<div class="line"><a name="l01164"></a><span class="lineno"> 1164</span>&#160;    <span class="comment">//</span></div>
<div class="line"><a name="l01165"></a><span class="lineno"> 1165</span>&#160;    <span class="comment">// The polynomial coefficients were computed by the Sollya script:</span></div>
<div class="line"><a name="l01166"></a><span class="lineno"> 1166</span>&#160;    <span class="comment">//</span></div>
<div class="line"><a name="l01167"></a><span class="lineno"> 1167</span>&#160;    <span class="comment">// c = 2 / log(2);</span></div>
<div class="line"><a name="l01168"></a><span class="lineno"> 1168</span>&#160;    <span class="comment">// trans = c * (x-1)/(x+1);</span></div>
<div class="line"><a name="l01169"></a><span class="lineno"> 1169</span>&#160;    <span class="comment">// itrans = (1+x/c)/(1-x/c);</span></div>
<div class="line"><a name="l01170"></a><span class="lineno"> 1170</span>&#160;    <span class="comment">// interval=[trans(sqrt(0.5)); trans(sqrt(2))];</span></div>
<div class="line"><a name="l01171"></a><span class="lineno"> 1171</span>&#160;    <span class="comment">// print(interval);</span></div>
<div class="line"><a name="l01172"></a><span class="lineno"> 1172</span>&#160;    <span class="comment">// f = log2(itrans(x));</span></div>
<div class="line"><a name="l01173"></a><span class="lineno"> 1173</span>&#160;    <span class="comment">// p=fpminimax(f,[|1,3,5,7,9,11,13,15,17|],[|1,DD,double...|],interval,relative,floating);</span></div>
<div class="line"><a name="l01174"></a><span class="lineno"> 1174</span>&#160;    <span class="keyword">const</span> Packet q12 = pset1&lt;Packet&gt;(2.87074255468000586e-9);</div>
<div class="line"><a name="l01175"></a><span class="lineno"> 1175</span>&#160;    <span class="keyword">const</span> Packet q10 = pset1&lt;Packet&gt;(2.38957980901884082e-8);</div>
<div class="line"><a name="l01176"></a><span class="lineno"> 1176</span>&#160;    <span class="keyword">const</span> Packet q8 = pset1&lt;Packet&gt;(2.31032094540014656e-7);</div>
<div class="line"><a name="l01177"></a><span class="lineno"> 1177</span>&#160;    <span class="keyword">const</span> Packet q6 = pset1&lt;Packet&gt;(2.27279857398537278e-6);</div>
<div class="line"><a name="l01178"></a><span class="lineno"> 1178</span>&#160;    <span class="keyword">const</span> Packet q4 = pset1&lt;Packet&gt;(2.31271023278625638e-5);</div>
<div class="line"><a name="l01179"></a><span class="lineno"> 1179</span>&#160;    <span class="keyword">const</span> Packet q2 = pset1&lt;Packet&gt;(2.47556738444535513e-4);</div>
<div class="line"><a name="l01180"></a><span class="lineno"> 1180</span>&#160;    <span class="keyword">const</span> Packet q0 = pset1&lt;Packet&gt;(2.88543873228900172e-3);</div>
<div class="line"><a name="l01181"></a><span class="lineno"> 1181</span>&#160;    <span class="keyword">const</span> Packet C_hi = pset1&lt;Packet&gt;(0.0400377511598501157);</div>
<div class="line"><a name="l01182"></a><span class="lineno"> 1182</span>&#160;    <span class="keyword">const</span> Packet C_lo = pset1&lt;Packet&gt;(-4.77726582251425391e-19);</div>
<div class="line"><a name="l01183"></a><span class="lineno"> 1183</span>&#160;    <span class="keyword">const</span> Packet one = pset1&lt;Packet&gt;(1.0);</div>
<div class="line"><a name="l01184"></a><span class="lineno"> 1184</span>&#160; </div>
<div class="line"><a name="l01185"></a><span class="lineno"> 1185</span>&#160;    <span class="keyword">const</span> Packet cst_2_log2e_hi = pset1&lt;Packet&gt;(2.88539008177792677);</div>
<div class="line"><a name="l01186"></a><span class="lineno"> 1186</span>&#160;    <span class="keyword">const</span> Packet cst_2_log2e_lo = pset1&lt;Packet&gt;(4.07660016854549667e-17);</div>
<div class="line"><a name="l01187"></a><span class="lineno"> 1187</span>&#160;    <span class="comment">// c * (x - 1)</span></div>
<div class="line"><a name="l01188"></a><span class="lineno"> 1188</span>&#160;    Packet num_hi, num_lo;</div>
<div class="line"><a name="l01189"></a><span class="lineno"> 1189</span>&#160;    twoprod(cst_2_log2e_hi, cst_2_log2e_lo, psub(x, one), num_hi, num_lo);</div>
<div class="line"><a name="l01190"></a><span class="lineno"> 1190</span>&#160;    <span class="comment">// TODO(rmlarsen): Investigate if using the division algorithm by</span></div>
<div class="line"><a name="l01191"></a><span class="lineno"> 1191</span>&#160;    <span class="comment">// Muller et al. is faster/more accurate.</span></div>
<div class="line"><a name="l01192"></a><span class="lineno"> 1192</span>&#160;    <span class="comment">// 1 / (x + 1)</span></div>
<div class="line"><a name="l01193"></a><span class="lineno"> 1193</span>&#160;    Packet denom_hi, denom_lo;</div>
<div class="line"><a name="l01194"></a><span class="lineno"> 1194</span>&#160;    doubleword_reciprocal(padd(x, one), denom_hi, denom_lo);</div>
<div class="line"><a name="l01195"></a><span class="lineno"> 1195</span>&#160;    <span class="comment">// r =  c * (x-1) / (x+1),</span></div>
<div class="line"><a name="l01196"></a><span class="lineno"> 1196</span>&#160;    Packet r_hi, r_lo;</div>
<div class="line"><a name="l01197"></a><span class="lineno"> 1197</span>&#160;    twoprod(num_hi, num_lo, denom_hi, denom_lo, r_hi, r_lo);</div>
<div class="line"><a name="l01198"></a><span class="lineno"> 1198</span>&#160;    <span class="comment">// r2 = r * r</span></div>
<div class="line"><a name="l01199"></a><span class="lineno"> 1199</span>&#160;    Packet r2_hi, r2_lo;</div>
<div class="line"><a name="l01200"></a><span class="lineno"> 1200</span>&#160;    twoprod(r_hi, r_lo, r_hi, r_lo, r2_hi, r2_lo);</div>
<div class="line"><a name="l01201"></a><span class="lineno"> 1201</span>&#160;    <span class="comment">// r4 = r2 * r2</span></div>
<div class="line"><a name="l01202"></a><span class="lineno"> 1202</span>&#160;    Packet r4_hi, r4_lo;</div>
<div class="line"><a name="l01203"></a><span class="lineno"> 1203</span>&#160;    twoprod(r2_hi, r2_lo, r2_hi, r2_lo, r4_hi, r4_lo);</div>
<div class="line"><a name="l01204"></a><span class="lineno"> 1204</span>&#160; </div>
<div class="line"><a name="l01205"></a><span class="lineno"> 1205</span>&#160;    <span class="comment">// Evaluate Q(r^2) in working precision. We evaluate it in two parts</span></div>
<div class="line"><a name="l01206"></a><span class="lineno"> 1206</span>&#160;    <span class="comment">// (even and odd in r^2) to improve instruction level parallelism.</span></div>
<div class="line"><a name="l01207"></a><span class="lineno"> 1207</span>&#160;    Packet q_even = pmadd(q12, r4_hi, q8);</div>
<div class="line"><a name="l01208"></a><span class="lineno"> 1208</span>&#160;    Packet q_odd = pmadd(q10, r4_hi, q6);</div>
<div class="line"><a name="l01209"></a><span class="lineno"> 1209</span>&#160;    q_even = pmadd(q_even, r4_hi, q4);</div>
<div class="line"><a name="l01210"></a><span class="lineno"> 1210</span>&#160;    q_odd = pmadd(q_odd, r4_hi, q2);</div>
<div class="line"><a name="l01211"></a><span class="lineno"> 1211</span>&#160;    q_even = pmadd(q_even, r4_hi, q0);</div>
<div class="line"><a name="l01212"></a><span class="lineno"> 1212</span>&#160;    Packet q = pmadd(q_odd, r2_hi, q_even);</div>
<div class="line"><a name="l01213"></a><span class="lineno"> 1213</span>&#160; </div>
<div class="line"><a name="l01214"></a><span class="lineno"> 1214</span>&#160;    <span class="comment">// Now evaluate the low order terms of P(x) in double word precision.</span></div>
<div class="line"><a name="l01215"></a><span class="lineno"> 1215</span>&#160;    <span class="comment">// In the following, due to the increasing magnitude of the coefficients</span></div>
<div class="line"><a name="l01216"></a><span class="lineno"> 1216</span>&#160;    <span class="comment">// and r being constrained to [-0.5, 0.5] we can use fast_twosum instead</span></div>
<div class="line"><a name="l01217"></a><span class="lineno"> 1217</span>&#160;    <span class="comment">// of the slower twosum.</span></div>
<div class="line"><a name="l01218"></a><span class="lineno"> 1218</span>&#160;    <span class="comment">// Q(r^2) * r^2</span></div>
<div class="line"><a name="l01219"></a><span class="lineno"> 1219</span>&#160;    Packet p_hi, p_lo;</div>
<div class="line"><a name="l01220"></a><span class="lineno"> 1220</span>&#160;    twoprod(r2_hi, r2_lo, q, p_hi, p_lo);</div>
<div class="line"><a name="l01221"></a><span class="lineno"> 1221</span>&#160;    <span class="comment">// Q(r^2) * r^2 + C</span></div>
<div class="line"><a name="l01222"></a><span class="lineno"> 1222</span>&#160;    Packet p1_hi, p1_lo;</div>
<div class="line"><a name="l01223"></a><span class="lineno"> 1223</span>&#160;    fast_twosum(C_hi, C_lo, p_hi, p_lo, p1_hi, p1_lo);</div>
<div class="line"><a name="l01224"></a><span class="lineno"> 1224</span>&#160;    <span class="comment">// (Q(r^2) * r^2 + C) * r^2</span></div>
<div class="line"><a name="l01225"></a><span class="lineno"> 1225</span>&#160;    Packet p2_hi, p2_lo;</div>
<div class="line"><a name="l01226"></a><span class="lineno"> 1226</span>&#160;    twoprod(r2_hi, r2_lo, p1_hi, p1_lo, p2_hi, p2_lo);</div>
<div class="line"><a name="l01227"></a><span class="lineno"> 1227</span>&#160;    <span class="comment">// ((Q(r^2) * r^2 + C) * r^2 + 1)</span></div>
<div class="line"><a name="l01228"></a><span class="lineno"> 1228</span>&#160;    Packet p3_hi, p3_lo;</div>
<div class="line"><a name="l01229"></a><span class="lineno"> 1229</span>&#160;    fast_twosum(one, p2_hi, p2_lo, p3_hi, p3_lo);</div>
<div class="line"><a name="l01230"></a><span class="lineno"> 1230</span>&#160; </div>
<div class="line"><a name="l01231"></a><span class="lineno"> 1231</span>&#160;    <span class="comment">// log(z) ~= ((Q(r^2) * r^2 + C) * r^2 + 1) * r</span></div>
<div class="line"><a name="l01232"></a><span class="lineno"> 1232</span>&#160;    twoprod(p3_hi, p3_lo, r_hi, r_lo, log2_x_hi, log2_x_lo);</div>
<div class="line"><a name="l01233"></a><span class="lineno"> 1233</span>&#160;  }</div>
<div class="line"><a name="l01234"></a><span class="lineno"> 1234</span>&#160;};</div>
<div class="line"><a name="l01235"></a><span class="lineno"> 1235</span>&#160; </div>
<div class="line"><a name="l01236"></a><span class="lineno"> 1236</span>&#160;<span class="comment">// This function computes exp2(x) (i.e. 2**x).</span></div>
<div class="line"><a name="l01237"></a><span class="lineno"> 1237</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> Scalar&gt;</div>
<div class="line"><a name="l01238"></a><span class="lineno"> 1238</span>&#160;<span class="keyword">struct </span>fast_accurate_exp2 {</div>
<div class="line"><a name="l01239"></a><span class="lineno"> 1239</span>&#160;  <span class="keyword">template</span> &lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l01240"></a><span class="lineno"> 1240</span>&#160;  EIGEN_STRONG_INLINE</div>
<div class="line"><a name="l01241"></a><span class="lineno"> 1241</span>&#160;  Packet operator()(<span class="keyword">const</span> Packet&amp; x) {</div>
<div class="line"><a name="l01242"></a><span class="lineno"> 1242</span>&#160;    <span class="comment">// TODO(rmlarsen): Add a pexp2 packetop.</span></div>
<div class="line"><a name="l01243"></a><span class="lineno"> 1243</span>&#160;    <span class="keywordflow">return</span> pexp(pmul(pset1&lt;Packet&gt;(Scalar(EIGEN_LN2)), x));</div>
<div class="line"><a name="l01244"></a><span class="lineno"> 1244</span>&#160;  }</div>
<div class="line"><a name="l01245"></a><span class="lineno"> 1245</span>&#160;};</div>
<div class="line"><a name="l01246"></a><span class="lineno"> 1246</span>&#160; </div>
<div class="line"><a name="l01247"></a><span class="lineno"> 1247</span>&#160;<span class="comment">// This specialization uses a faster algorithm to compute exp2(x) for floats</span></div>
<div class="line"><a name="l01248"></a><span class="lineno"> 1248</span>&#160;<span class="comment">// in [-0.5;0.5] with a relative accuracy of 1 ulp.</span></div>
<div class="line"><a name="l01249"></a><span class="lineno"> 1249</span>&#160;<span class="comment">// The minimax polynomial used was calculated using the Sollya tool.</span></div>
<div class="line"><a name="l01250"></a><span class="lineno"> 1250</span>&#160;<span class="comment">// See sollya.org.</span></div>
<div class="line"><a name="l01251"></a><span class="lineno"> 1251</span>&#160;<span class="keyword">template</span> &lt;&gt;</div>
<div class="line"><a name="l01252"></a><span class="lineno"> 1252</span>&#160;<span class="keyword">struct </span>fast_accurate_exp2&lt;float&gt; {</div>
<div class="line"><a name="l01253"></a><span class="lineno"> 1253</span>&#160;  <span class="keyword">template</span> &lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l01254"></a><span class="lineno"> 1254</span>&#160;  EIGEN_STRONG_INLINE</div>
<div class="line"><a name="l01255"></a><span class="lineno"> 1255</span>&#160;  Packet operator()(<span class="keyword">const</span> Packet&amp; x) {</div>
<div class="line"><a name="l01256"></a><span class="lineno"> 1256</span>&#160;    <span class="comment">// This function approximates exp2(x) by a degree 6 polynomial of the form</span></div>
<div class="line"><a name="l01257"></a><span class="lineno"> 1257</span>&#160;    <span class="comment">// Q(x) = 1 + x * (C + x * P(x)), where the degree 4 polynomial P(x) is evaluated in</span></div>
<div class="line"><a name="l01258"></a><span class="lineno"> 1258</span>&#160;    <span class="comment">// single precision, and the remaining steps are evaluated with extra precision using</span></div>
<div class="line"><a name="l01259"></a><span class="lineno"> 1259</span>&#160;    <span class="comment">// double word arithmetic. C is an extra precise constant stored as a double word.</span></div>
<div class="line"><a name="l01260"></a><span class="lineno"> 1260</span>&#160;    <span class="comment">//</span></div>
<div class="line"><a name="l01261"></a><span class="lineno"> 1261</span>&#160;    <span class="comment">// The polynomial coefficients were calculated using Sollya commands:</span></div>
<div class="line"><a name="l01262"></a><span class="lineno"> 1262</span>&#160;    <span class="comment">// &gt; n = 6;</span></div>
<div class="line"><a name="l01263"></a><span class="lineno"> 1263</span>&#160;    <span class="comment">// &gt; f = 2^x;</span></div>
<div class="line"><a name="l01264"></a><span class="lineno"> 1264</span>&#160;    <span class="comment">// &gt; interval = [-0.5;0.5];</span></div>
<div class="line"><a name="l01265"></a><span class="lineno"> 1265</span>&#160;    <span class="comment">// &gt; p = fpminimax(f,n,[|1,double,single...|],interval,relative,floating);</span></div>
<div class="line"><a name="l01266"></a><span class="lineno"> 1266</span>&#160; </div>
<div class="line"><a name="l01267"></a><span class="lineno"> 1267</span>&#160;    <span class="keyword">const</span> Packet p4 = pset1&lt;Packet&gt;(1.539513905e-4f);</div>
<div class="line"><a name="l01268"></a><span class="lineno"> 1268</span>&#160;    <span class="keyword">const</span> Packet p3 = pset1&lt;Packet&gt;(1.340007293e-3f);</div>
<div class="line"><a name="l01269"></a><span class="lineno"> 1269</span>&#160;    <span class="keyword">const</span> Packet p2 = pset1&lt;Packet&gt;(9.618283249e-3f);</div>
<div class="line"><a name="l01270"></a><span class="lineno"> 1270</span>&#160;    <span class="keyword">const</span> Packet p1 = pset1&lt;Packet&gt;(5.550328270e-2f);</div>
<div class="line"><a name="l01271"></a><span class="lineno"> 1271</span>&#160;    <span class="keyword">const</span> Packet p0 = pset1&lt;Packet&gt;(0.2402264923f);</div>
<div class="line"><a name="l01272"></a><span class="lineno"> 1272</span>&#160; </div>
<div class="line"><a name="l01273"></a><span class="lineno"> 1273</span>&#160;    <span class="keyword">const</span> Packet C_hi = pset1&lt;Packet&gt;(0.6931471825f);</div>
<div class="line"><a name="l01274"></a><span class="lineno"> 1274</span>&#160;    <span class="keyword">const</span> Packet C_lo = pset1&lt;Packet&gt;(2.36836577e-08f);</div>
<div class="line"><a name="l01275"></a><span class="lineno"> 1275</span>&#160;    <span class="keyword">const</span> Packet one = pset1&lt;Packet&gt;(1.0f);</div>
<div class="line"><a name="l01276"></a><span class="lineno"> 1276</span>&#160; </div>
<div class="line"><a name="l01277"></a><span class="lineno"> 1277</span>&#160;    <span class="comment">// Evaluate P(x) in working precision.</span></div>
<div class="line"><a name="l01278"></a><span class="lineno"> 1278</span>&#160;    <span class="comment">// We evaluate even and odd parts of the polynomial separately</span></div>
<div class="line"><a name="l01279"></a><span class="lineno"> 1279</span>&#160;    <span class="comment">// to gain some instruction level parallelism.</span></div>
<div class="line"><a name="l01280"></a><span class="lineno"> 1280</span>&#160;    Packet x2 = pmul(x,x);</div>
<div class="line"><a name="l01281"></a><span class="lineno"> 1281</span>&#160;    Packet p_even = pmadd(p4, x2, p2);</div>
<div class="line"><a name="l01282"></a><span class="lineno"> 1282</span>&#160;    Packet p_odd = pmadd(p3, x2, p1);</div>
<div class="line"><a name="l01283"></a><span class="lineno"> 1283</span>&#160;    p_even = pmadd(p_even, x2, p0);</div>
<div class="line"><a name="l01284"></a><span class="lineno"> 1284</span>&#160;    Packet p = pmadd(p_odd, x, p_even);</div>
<div class="line"><a name="l01285"></a><span class="lineno"> 1285</span>&#160; </div>
<div class="line"><a name="l01286"></a><span class="lineno"> 1286</span>&#160;    <span class="comment">// Evaluate the remaining terms of Q(x) with extra precision using</span></div>
<div class="line"><a name="l01287"></a><span class="lineno"> 1287</span>&#160;    <span class="comment">// double word arithmetic.</span></div>
<div class="line"><a name="l01288"></a><span class="lineno"> 1288</span>&#160;    Packet p_hi, p_lo;</div>
<div class="line"><a name="l01289"></a><span class="lineno"> 1289</span>&#160;    <span class="comment">// x * p(x)</span></div>
<div class="line"><a name="l01290"></a><span class="lineno"> 1290</span>&#160;    twoprod(p, x, p_hi, p_lo);</div>
<div class="line"><a name="l01291"></a><span class="lineno"> 1291</span>&#160;    <span class="comment">// C + x * p(x)</span></div>
<div class="line"><a name="l01292"></a><span class="lineno"> 1292</span>&#160;    Packet q1_hi, q1_lo;</div>
<div class="line"><a name="l01293"></a><span class="lineno"> 1293</span>&#160;    twosum(p_hi, p_lo, C_hi, C_lo, q1_hi, q1_lo);</div>
<div class="line"><a name="l01294"></a><span class="lineno"> 1294</span>&#160;    <span class="comment">// x * (C + x * p(x))</span></div>
<div class="line"><a name="l01295"></a><span class="lineno"> 1295</span>&#160;    Packet q2_hi, q2_lo;</div>
<div class="line"><a name="l01296"></a><span class="lineno"> 1296</span>&#160;    twoprod(q1_hi, q1_lo, x, q2_hi, q2_lo);</div>
<div class="line"><a name="l01297"></a><span class="lineno"> 1297</span>&#160;    <span class="comment">// 1 + x * (C + x * p(x))</span></div>
<div class="line"><a name="l01298"></a><span class="lineno"> 1298</span>&#160;    Packet q3_hi, q3_lo;</div>
<div class="line"><a name="l01299"></a><span class="lineno"> 1299</span>&#160;    <span class="comment">// Since |q2_hi| &lt;= sqrt(2)-1 &lt; 1, we can use fast_twosum</span></div>
<div class="line"><a name="l01300"></a><span class="lineno"> 1300</span>&#160;    <span class="comment">// for adding it to unity here.</span></div>
<div class="line"><a name="l01301"></a><span class="lineno"> 1301</span>&#160;    fast_twosum(one, q2_hi, q3_hi, q3_lo);</div>
<div class="line"><a name="l01302"></a><span class="lineno"> 1302</span>&#160;    <span class="keywordflow">return</span> padd(q3_hi, padd(q2_lo, q3_lo));</div>
<div class="line"><a name="l01303"></a><span class="lineno"> 1303</span>&#160;  }</div>
<div class="line"><a name="l01304"></a><span class="lineno"> 1304</span>&#160;};</div>
<div class="line"><a name="l01305"></a><span class="lineno"> 1305</span>&#160; </div>
<div class="line"><a name="l01306"></a><span class="lineno"> 1306</span>&#160;<span class="comment">// in [-0.5;0.5] with a relative accuracy of 1 ulp.</span></div>
<div class="line"><a name="l01307"></a><span class="lineno"> 1307</span>&#160;<span class="comment">// The minimax polynomial used was calculated using the Sollya tool.</span></div>
<div class="line"><a name="l01308"></a><span class="lineno"> 1308</span>&#160;<span class="comment">// See sollya.org.</span></div>
<div class="line"><a name="l01309"></a><span class="lineno"> 1309</span>&#160;<span class="keyword">template</span> &lt;&gt;</div>
<div class="line"><a name="l01310"></a><span class="lineno"> 1310</span>&#160;<span class="keyword">struct </span>fast_accurate_exp2&lt;double&gt; {</div>
<div class="line"><a name="l01311"></a><span class="lineno"> 1311</span>&#160;  <span class="keyword">template</span> &lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l01312"></a><span class="lineno"> 1312</span>&#160;  EIGEN_STRONG_INLINE</div>
<div class="line"><a name="l01313"></a><span class="lineno"> 1313</span>&#160;  Packet operator()(<span class="keyword">const</span> Packet&amp; x) {</div>
<div class="line"><a name="l01314"></a><span class="lineno"> 1314</span>&#160;    <span class="comment">// This function approximates exp2(x) by a degree 10 polynomial of the form</span></div>
<div class="line"><a name="l01315"></a><span class="lineno"> 1315</span>&#160;    <span class="comment">// Q(x) = 1 + x * (C + x * P(x)), where the degree 8 polynomial P(x) is evaluated in</span></div>
<div class="line"><a name="l01316"></a><span class="lineno"> 1316</span>&#160;    <span class="comment">// single precision, and the remaining steps are evaluated with extra precision using</span></div>
<div class="line"><a name="l01317"></a><span class="lineno"> 1317</span>&#160;    <span class="comment">// double word arithmetic. C is an extra precise constant stored as a double word.</span></div>
<div class="line"><a name="l01318"></a><span class="lineno"> 1318</span>&#160;    <span class="comment">//</span></div>
<div class="line"><a name="l01319"></a><span class="lineno"> 1319</span>&#160;    <span class="comment">// The polynomial coefficients were calculated using Sollya commands:</span></div>
<div class="line"><a name="l01320"></a><span class="lineno"> 1320</span>&#160;    <span class="comment">// &gt; n = 11;</span></div>
<div class="line"><a name="l01321"></a><span class="lineno"> 1321</span>&#160;    <span class="comment">// &gt; f = 2^x;</span></div>
<div class="line"><a name="l01322"></a><span class="lineno"> 1322</span>&#160;    <span class="comment">// &gt; interval = [-0.5;0.5];</span></div>
<div class="line"><a name="l01323"></a><span class="lineno"> 1323</span>&#160;    <span class="comment">// &gt; p = fpminimax(f,n,[|1,DD,double...|],interval,relative,floating);</span></div>
<div class="line"><a name="l01324"></a><span class="lineno"> 1324</span>&#160; </div>
<div class="line"><a name="l01325"></a><span class="lineno"> 1325</span>&#160;    <span class="keyword">const</span> Packet p9 = pset1&lt;Packet&gt;(4.431642109085495276e-10);</div>
<div class="line"><a name="l01326"></a><span class="lineno"> 1326</span>&#160;    <span class="keyword">const</span> Packet p8 = pset1&lt;Packet&gt;(7.073829923303358410e-9);</div>
<div class="line"><a name="l01327"></a><span class="lineno"> 1327</span>&#160;    <span class="keyword">const</span> Packet p7 = pset1&lt;Packet&gt;(1.017822306737031311e-7);</div>
<div class="line"><a name="l01328"></a><span class="lineno"> 1328</span>&#160;    <span class="keyword">const</span> Packet p6 = pset1&lt;Packet&gt;(1.321543498017646657e-6);</div>
<div class="line"><a name="l01329"></a><span class="lineno"> 1329</span>&#160;    <span class="keyword">const</span> Packet p5 = pset1&lt;Packet&gt;(1.525273342728892877e-5);</div>
<div class="line"><a name="l01330"></a><span class="lineno"> 1330</span>&#160;    <span class="keyword">const</span> Packet p4 = pset1&lt;Packet&gt;(1.540353045780084423e-4);</div>
<div class="line"><a name="l01331"></a><span class="lineno"> 1331</span>&#160;    <span class="keyword">const</span> Packet p3 = pset1&lt;Packet&gt;(1.333355814685869807e-3);</div>
<div class="line"><a name="l01332"></a><span class="lineno"> 1332</span>&#160;    <span class="keyword">const</span> Packet p2 = pset1&lt;Packet&gt;(9.618129107593478832e-3);</div>
<div class="line"><a name="l01333"></a><span class="lineno"> 1333</span>&#160;    <span class="keyword">const</span> Packet p1 = pset1&lt;Packet&gt;(5.550410866481961247e-2);</div>
<div class="line"><a name="l01334"></a><span class="lineno"> 1334</span>&#160;    <span class="keyword">const</span> Packet p0 = pset1&lt;Packet&gt;(0.240226506959101332);</div>
<div class="line"><a name="l01335"></a><span class="lineno"> 1335</span>&#160;    <span class="keyword">const</span> Packet C_hi = pset1&lt;Packet&gt;(0.693147180559945286); </div>
<div class="line"><a name="l01336"></a><span class="lineno"> 1336</span>&#160;    <span class="keyword">const</span> Packet C_lo = pset1&lt;Packet&gt;(4.81927865669806721e-17);</div>
<div class="line"><a name="l01337"></a><span class="lineno"> 1337</span>&#160;    <span class="keyword">const</span> Packet one = pset1&lt;Packet&gt;(1.0);</div>
<div class="line"><a name="l01338"></a><span class="lineno"> 1338</span>&#160; </div>
<div class="line"><a name="l01339"></a><span class="lineno"> 1339</span>&#160;    <span class="comment">// Evaluate P(x) in working precision.</span></div>
<div class="line"><a name="l01340"></a><span class="lineno"> 1340</span>&#160;    <span class="comment">// We evaluate even and odd parts of the polynomial separately</span></div>
<div class="line"><a name="l01341"></a><span class="lineno"> 1341</span>&#160;    <span class="comment">// to gain some instruction level parallelism.</span></div>
<div class="line"><a name="l01342"></a><span class="lineno"> 1342</span>&#160;    Packet x2 = pmul(x,x);</div>
<div class="line"><a name="l01343"></a><span class="lineno"> 1343</span>&#160;    Packet p_even = pmadd(p8, x2, p6);</div>
<div class="line"><a name="l01344"></a><span class="lineno"> 1344</span>&#160;    Packet p_odd = pmadd(p9, x2, p7);</div>
<div class="line"><a name="l01345"></a><span class="lineno"> 1345</span>&#160;    p_even = pmadd(p_even, x2, p4);</div>
<div class="line"><a name="l01346"></a><span class="lineno"> 1346</span>&#160;    p_odd = pmadd(p_odd, x2, p5);</div>
<div class="line"><a name="l01347"></a><span class="lineno"> 1347</span>&#160;    p_even = pmadd(p_even, x2, p2);</div>
<div class="line"><a name="l01348"></a><span class="lineno"> 1348</span>&#160;    p_odd = pmadd(p_odd, x2, p3);</div>
<div class="line"><a name="l01349"></a><span class="lineno"> 1349</span>&#160;    p_even = pmadd(p_even, x2, p0);</div>
<div class="line"><a name="l01350"></a><span class="lineno"> 1350</span>&#160;    p_odd = pmadd(p_odd, x2, p1);</div>
<div class="line"><a name="l01351"></a><span class="lineno"> 1351</span>&#160;    Packet p = pmadd(p_odd, x, p_even);</div>
<div class="line"><a name="l01352"></a><span class="lineno"> 1352</span>&#160; </div>
<div class="line"><a name="l01353"></a><span class="lineno"> 1353</span>&#160;    <span class="comment">// Evaluate the remaining terms of Q(x) with extra precision using</span></div>
<div class="line"><a name="l01354"></a><span class="lineno"> 1354</span>&#160;    <span class="comment">// double word arithmetic.</span></div>
<div class="line"><a name="l01355"></a><span class="lineno"> 1355</span>&#160;    Packet p_hi, p_lo;</div>
<div class="line"><a name="l01356"></a><span class="lineno"> 1356</span>&#160;    <span class="comment">// x * p(x)</span></div>
<div class="line"><a name="l01357"></a><span class="lineno"> 1357</span>&#160;    twoprod(p, x, p_hi, p_lo);</div>
<div class="line"><a name="l01358"></a><span class="lineno"> 1358</span>&#160;    <span class="comment">// C + x * p(x)</span></div>
<div class="line"><a name="l01359"></a><span class="lineno"> 1359</span>&#160;    Packet q1_hi, q1_lo;</div>
<div class="line"><a name="l01360"></a><span class="lineno"> 1360</span>&#160;    twosum(p_hi, p_lo, C_hi, C_lo, q1_hi, q1_lo);</div>
<div class="line"><a name="l01361"></a><span class="lineno"> 1361</span>&#160;    <span class="comment">// x * (C + x * p(x))</span></div>
<div class="line"><a name="l01362"></a><span class="lineno"> 1362</span>&#160;    Packet q2_hi, q2_lo;</div>
<div class="line"><a name="l01363"></a><span class="lineno"> 1363</span>&#160;    twoprod(q1_hi, q1_lo, x, q2_hi, q2_lo);</div>
<div class="line"><a name="l01364"></a><span class="lineno"> 1364</span>&#160;    <span class="comment">// 1 + x * (C + x * p(x))</span></div>
<div class="line"><a name="l01365"></a><span class="lineno"> 1365</span>&#160;    Packet q3_hi, q3_lo;</div>
<div class="line"><a name="l01366"></a><span class="lineno"> 1366</span>&#160;    <span class="comment">// Since |q2_hi| &lt;= sqrt(2)-1 &lt; 1, we can use fast_twosum</span></div>
<div class="line"><a name="l01367"></a><span class="lineno"> 1367</span>&#160;    <span class="comment">// for adding it to unity here.</span></div>
<div class="line"><a name="l01368"></a><span class="lineno"> 1368</span>&#160;    fast_twosum(one, q2_hi, q3_hi, q3_lo);</div>
<div class="line"><a name="l01369"></a><span class="lineno"> 1369</span>&#160;    <span class="keywordflow">return</span> padd(q3_hi, padd(q2_lo, q3_lo));</div>
<div class="line"><a name="l01370"></a><span class="lineno"> 1370</span>&#160;  }</div>
<div class="line"><a name="l01371"></a><span class="lineno"> 1371</span>&#160;};</div>
<div class="line"><a name="l01372"></a><span class="lineno"> 1372</span>&#160; </div>
<div class="line"><a name="l01373"></a><span class="lineno"> 1373</span>&#160;<span class="comment">// This function implements the non-trivial case of pow(x,y) where x is</span></div>
<div class="line"><a name="l01374"></a><span class="lineno"> 1374</span>&#160;<span class="comment">// positive and y is (possibly) non-integer.</span></div>
<div class="line"><a name="l01375"></a><span class="lineno"> 1375</span>&#160;<span class="comment">// Formally, pow(x,y) = exp2(y * log2(x)), where exp2(x) is shorthand for 2^x.</span></div>
<div class="line"><a name="l01376"></a><span class="lineno"> 1376</span>&#160;<span class="comment">// TODO(rmlarsen): We should probably add this as a packet up &#39;ppow&#39;, to make it</span></div>
<div class="line"><a name="l01377"></a><span class="lineno"> 1377</span>&#160;<span class="comment">// easier to specialize or turn off for specific types and/or backends.x</span></div>
<div class="line"><a name="l01378"></a><span class="lineno"> 1378</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l01379"></a><span class="lineno"> 1379</span>&#160;EIGEN_STRONG_INLINE Packet generic_pow_impl(<span class="keyword">const</span> Packet&amp; x, <span class="keyword">const</span> Packet&amp; y) {</div>
<div class="line"><a name="l01380"></a><span class="lineno"> 1380</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::type Scalar;</div>
<div class="line"><a name="l01381"></a><span class="lineno"> 1381</span>&#160;  <span class="comment">// Split x into exponent e_x and mantissa m_x.</span></div>
<div class="line"><a name="l01382"></a><span class="lineno"> 1382</span>&#160;  Packet e_x;</div>
<div class="line"><a name="l01383"></a><span class="lineno"> 1383</span>&#160;  Packet m_x = pfrexp(x, e_x);</div>
<div class="line"><a name="l01384"></a><span class="lineno"> 1384</span>&#160; </div>
<div class="line"><a name="l01385"></a><span class="lineno"> 1385</span>&#160;  <span class="comment">// Adjust m_x to lie in [1/sqrt(2):sqrt(2)] to minimize absolute error in log2(m_x).</span></div>
<div class="line"><a name="l01386"></a><span class="lineno"> 1386</span>&#160;  EIGEN_CONSTEXPR Scalar sqrt_half = Scalar(0.70710678118654752440);</div>
<div class="line"><a name="l01387"></a><span class="lineno"> 1387</span>&#160;  <span class="keyword">const</span> Packet m_x_scale_mask = pcmp_lt(m_x, pset1&lt;Packet&gt;(sqrt_half));</div>
<div class="line"><a name="l01388"></a><span class="lineno"> 1388</span>&#160;  m_x = pselect(m_x_scale_mask, pmul(pset1&lt;Packet&gt;(Scalar(2)), m_x), m_x);</div>
<div class="line"><a name="l01389"></a><span class="lineno"> 1389</span>&#160;  e_x = pselect(m_x_scale_mask, psub(e_x, pset1&lt;Packet&gt;(Scalar(1))), e_x);</div>
<div class="line"><a name="l01390"></a><span class="lineno"> 1390</span>&#160; </div>
<div class="line"><a name="l01391"></a><span class="lineno"> 1391</span>&#160;  <span class="comment">// Compute log2(m_x) with 6 extra bits of accuracy.</span></div>
<div class="line"><a name="l01392"></a><span class="lineno"> 1392</span>&#160;  Packet rx_hi, rx_lo;</div>
<div class="line"><a name="l01393"></a><span class="lineno"> 1393</span>&#160;  accurate_log2&lt;Scalar&gt;()(m_x, rx_hi, rx_lo);</div>
<div class="line"><a name="l01394"></a><span class="lineno"> 1394</span>&#160; </div>
<div class="line"><a name="l01395"></a><span class="lineno"> 1395</span>&#160;  <span class="comment">// Compute the two terms {y * e_x, y * r_x} in f = y * log2(x) with doubled</span></div>
<div class="line"><a name="l01396"></a><span class="lineno"> 1396</span>&#160;  <span class="comment">// precision using double word arithmetic.</span></div>
<div class="line"><a name="l01397"></a><span class="lineno"> 1397</span>&#160;  Packet f1_hi, f1_lo, f2_hi, f2_lo;</div>
<div class="line"><a name="l01398"></a><span class="lineno"> 1398</span>&#160;  twoprod(e_x, y, f1_hi, f1_lo);</div>
<div class="line"><a name="l01399"></a><span class="lineno"> 1399</span>&#160;  twoprod(rx_hi, rx_lo, y, f2_hi, f2_lo);</div>
<div class="line"><a name="l01400"></a><span class="lineno"> 1400</span>&#160;  <span class="comment">// Sum the two terms in f using double word arithmetic. We know</span></div>
<div class="line"><a name="l01401"></a><span class="lineno"> 1401</span>&#160;  <span class="comment">// that |e_x| &gt; |log2(m_x)|, except for the case where e_x==0.</span></div>
<div class="line"><a name="l01402"></a><span class="lineno"> 1402</span>&#160;  <span class="comment">// This means that we can use fast_twosum(f1,f2).</span></div>
<div class="line"><a name="l01403"></a><span class="lineno"> 1403</span>&#160;  <span class="comment">// In the case e_x == 0, e_x * y = f1 = 0, so we don&#39;t lose any</span></div>
<div class="line"><a name="l01404"></a><span class="lineno"> 1404</span>&#160;  <span class="comment">// accuracy by violating the assumption of fast_twosum, because</span></div>
<div class="line"><a name="l01405"></a><span class="lineno"> 1405</span>&#160;  <span class="comment">// it&#39;s a no-op.</span></div>
<div class="line"><a name="l01406"></a><span class="lineno"> 1406</span>&#160;  Packet f_hi, f_lo;</div>
<div class="line"><a name="l01407"></a><span class="lineno"> 1407</span>&#160;  fast_twosum(f1_hi, f1_lo, f2_hi, f2_lo, f_hi, f_lo);</div>
<div class="line"><a name="l01408"></a><span class="lineno"> 1408</span>&#160; </div>
<div class="line"><a name="l01409"></a><span class="lineno"> 1409</span>&#160;  <span class="comment">// Split f into integer and fractional parts.</span></div>
<div class="line"><a name="l01410"></a><span class="lineno"> 1410</span>&#160;  Packet n_z, r_z;</div>
<div class="line"><a name="l01411"></a><span class="lineno"> 1411</span>&#160;  absolute_split(f_hi, n_z, r_z);</div>
<div class="line"><a name="l01412"></a><span class="lineno"> 1412</span>&#160;  r_z = padd(r_z, f_lo);</div>
<div class="line"><a name="l01413"></a><span class="lineno"> 1413</span>&#160;  Packet n_r;</div>
<div class="line"><a name="l01414"></a><span class="lineno"> 1414</span>&#160;  absolute_split(r_z, n_r, r_z);</div>
<div class="line"><a name="l01415"></a><span class="lineno"> 1415</span>&#160;  n_z = padd(n_z, n_r);</div>
<div class="line"><a name="l01416"></a><span class="lineno"> 1416</span>&#160; </div>
<div class="line"><a name="l01417"></a><span class="lineno"> 1417</span>&#160;  <span class="comment">// We now have an accurate split of f = n_z + r_z and can compute</span></div>
<div class="line"><a name="l01418"></a><span class="lineno"> 1418</span>&#160;  <span class="comment">//   x^y = 2**{n_z + r_z) = exp2(r_z) * 2**{n_z}.</span></div>
<div class="line"><a name="l01419"></a><span class="lineno"> 1419</span>&#160;  <span class="comment">// Since r_z is in [-0.5;0.5], we compute the first factor to high accuracy</span></div>
<div class="line"><a name="l01420"></a><span class="lineno"> 1420</span>&#160;  <span class="comment">// using a specialized algorithm. Multiplication by the second factor can</span></div>
<div class="line"><a name="l01421"></a><span class="lineno"> 1421</span>&#160;  <span class="comment">// be done exactly using pldexp(), since it is an integer power of 2.</span></div>
<div class="line"><a name="l01422"></a><span class="lineno"> 1422</span>&#160;  <span class="keyword">const</span> Packet e_r = fast_accurate_exp2&lt;Scalar&gt;()(r_z);</div>
<div class="line"><a name="l01423"></a><span class="lineno"> 1423</span>&#160;  <span class="keywordflow">return</span> pldexp(e_r, n_z);</div>
<div class="line"><a name="l01424"></a><span class="lineno"> 1424</span>&#160;}</div>
<div class="line"><a name="l01425"></a><span class="lineno"> 1425</span>&#160; </div>
<div class="line"><a name="l01426"></a><span class="lineno"> 1426</span>&#160;<span class="comment">// Generic implementation of pow(x,y).</span></div>
<div class="line"><a name="l01427"></a><span class="lineno"> 1427</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l01428"></a><span class="lineno"> 1428</span>&#160;EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS</div>
<div class="line"><a name="l01429"></a><span class="lineno"> 1429</span>&#160;Packet generic_pow(<span class="keyword">const</span> Packet&amp; x, <span class="keyword">const</span> Packet&amp; y) {</div>
<div class="line"><a name="l01430"></a><span class="lineno"> 1430</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::type Scalar;</div>
<div class="line"><a name="l01431"></a><span class="lineno"> 1431</span>&#160; </div>
<div class="line"><a name="l01432"></a><span class="lineno"> 1432</span>&#160;  <span class="keyword">const</span> Packet cst_pos_inf = pset1&lt;Packet&gt;(NumTraits&lt;Scalar&gt;::infinity());</div>
<div class="line"><a name="l01433"></a><span class="lineno"> 1433</span>&#160;  <span class="keyword">const</span> Packet cst_zero = pset1&lt;Packet&gt;(Scalar(0));</div>
<div class="line"><a name="l01434"></a><span class="lineno"> 1434</span>&#160;  <span class="keyword">const</span> Packet cst_one = pset1&lt;Packet&gt;(Scalar(1));</div>
<div class="line"><a name="l01435"></a><span class="lineno"> 1435</span>&#160;  <span class="keyword">const</span> Packet cst_nan = pset1&lt;Packet&gt;(NumTraits&lt;Scalar&gt;::quiet_NaN());</div>
<div class="line"><a name="l01436"></a><span class="lineno"> 1436</span>&#160; </div>
<div class="line"><a name="l01437"></a><span class="lineno"> 1437</span>&#160;  <span class="keyword">const</span> Packet abs_x = pabs(x);</div>
<div class="line"><a name="l01438"></a><span class="lineno"> 1438</span>&#160;  <span class="comment">// Predicates for sign and magnitude of x.</span></div>
<div class="line"><a name="l01439"></a><span class="lineno"> 1439</span>&#160;  <span class="keyword">const</span> Packet x_is_zero = pcmp_eq(x, cst_zero);</div>
<div class="line"><a name="l01440"></a><span class="lineno"> 1440</span>&#160;  <span class="keyword">const</span> Packet x_is_neg = pcmp_lt(x, cst_zero);</div>
<div class="line"><a name="l01441"></a><span class="lineno"> 1441</span>&#160;  <span class="keyword">const</span> Packet abs_x_is_inf = pcmp_eq(abs_x, cst_pos_inf);</div>
<div class="line"><a name="l01442"></a><span class="lineno"> 1442</span>&#160;  <span class="keyword">const</span> Packet abs_x_is_one =  pcmp_eq(abs_x, cst_one);</div>
<div class="line"><a name="l01443"></a><span class="lineno"> 1443</span>&#160;  <span class="keyword">const</span> Packet abs_x_is_gt_one = pcmp_lt(cst_one, abs_x);</div>
<div class="line"><a name="l01444"></a><span class="lineno"> 1444</span>&#160;  <span class="keyword">const</span> Packet abs_x_is_lt_one = pcmp_lt(abs_x, cst_one);</div>
<div class="line"><a name="l01445"></a><span class="lineno"> 1445</span>&#160;  <span class="keyword">const</span> Packet x_is_one =  pandnot(abs_x_is_one, x_is_neg);</div>
<div class="line"><a name="l01446"></a><span class="lineno"> 1446</span>&#160;  <span class="keyword">const</span> Packet x_is_neg_one =  pand(abs_x_is_one, x_is_neg);</div>
<div class="line"><a name="l01447"></a><span class="lineno"> 1447</span>&#160;  <span class="keyword">const</span> Packet x_is_nan = pandnot(ptrue(x), pcmp_eq(x, x));</div>
<div class="line"><a name="l01448"></a><span class="lineno"> 1448</span>&#160; </div>
<div class="line"><a name="l01449"></a><span class="lineno"> 1449</span>&#160;  <span class="comment">// Predicates for sign and magnitude of y.</span></div>
<div class="line"><a name="l01450"></a><span class="lineno"> 1450</span>&#160;  <span class="keyword">const</span> Packet y_is_one = pcmp_eq(y, cst_one);</div>
<div class="line"><a name="l01451"></a><span class="lineno"> 1451</span>&#160;  <span class="keyword">const</span> Packet y_is_zero = pcmp_eq(y, cst_zero);</div>
<div class="line"><a name="l01452"></a><span class="lineno"> 1452</span>&#160;  <span class="keyword">const</span> Packet y_is_neg = pcmp_lt(y, cst_zero);</div>
<div class="line"><a name="l01453"></a><span class="lineno"> 1453</span>&#160;  <span class="keyword">const</span> Packet y_is_pos = pandnot(ptrue(y), por(y_is_zero, y_is_neg));</div>
<div class="line"><a name="l01454"></a><span class="lineno"> 1454</span>&#160;  <span class="keyword">const</span> Packet y_is_nan = pandnot(ptrue(y), pcmp_eq(y, y));</div>
<div class="line"><a name="l01455"></a><span class="lineno"> 1455</span>&#160;  <span class="keyword">const</span> Packet abs_y_is_inf = pcmp_eq(pabs(y), cst_pos_inf);</div>
<div class="line"><a name="l01456"></a><span class="lineno"> 1456</span>&#160;  EIGEN_CONSTEXPR Scalar huge_exponent =</div>
<div class="line"><a name="l01457"></a><span class="lineno"> 1457</span>&#160;      (NumTraits&lt;Scalar&gt;::max_exponent() * Scalar(EIGEN_LN2)) /</div>
<div class="line"><a name="l01458"></a><span class="lineno"> 1458</span>&#160;       NumTraits&lt;Scalar&gt;::epsilon();</div>
<div class="line"><a name="l01459"></a><span class="lineno"> 1459</span>&#160;  <span class="keyword">const</span> Packet abs_y_is_huge = pcmp_le(pset1&lt;Packet&gt;(huge_exponent), pabs(y));</div>
<div class="line"><a name="l01460"></a><span class="lineno"> 1460</span>&#160; </div>
<div class="line"><a name="l01461"></a><span class="lineno"> 1461</span>&#160;  <span class="comment">// Predicates for whether y is integer and/or even.</span></div>
<div class="line"><a name="l01462"></a><span class="lineno"> 1462</span>&#160;  <span class="keyword">const</span> Packet y_is_int = pcmp_eq(pfloor(y), y);</div>
<div class="line"><a name="l01463"></a><span class="lineno"> 1463</span>&#160;  <span class="keyword">const</span> Packet y_div_2 = pmul(y, pset1&lt;Packet&gt;(Scalar(0.5)));</div>
<div class="line"><a name="l01464"></a><span class="lineno"> 1464</span>&#160;  <span class="keyword">const</span> Packet y_is_even = pcmp_eq(pround(y_div_2), y_div_2);</div>
<div class="line"><a name="l01465"></a><span class="lineno"> 1465</span>&#160; </div>
<div class="line"><a name="l01466"></a><span class="lineno"> 1466</span>&#160;  <span class="comment">// Predicates encoding special cases for the value of pow(x,y)</span></div>
<div class="line"><a name="l01467"></a><span class="lineno"> 1467</span>&#160;  <span class="keyword">const</span> Packet invalid_negative_x = pandnot(pandnot(pandnot(x_is_neg, abs_x_is_inf),</div>
<div class="line"><a name="l01468"></a><span class="lineno"> 1468</span>&#160;                                                    y_is_int),</div>
<div class="line"><a name="l01469"></a><span class="lineno"> 1469</span>&#160;                                            abs_y_is_inf);</div>
<div class="line"><a name="l01470"></a><span class="lineno"> 1470</span>&#160;  <span class="keyword">const</span> Packet pow_is_one = por(por(x_is_one, y_is_zero),</div>
<div class="line"><a name="l01471"></a><span class="lineno"> 1471</span>&#160;                                pand(x_is_neg_one,</div>
<div class="line"><a name="l01472"></a><span class="lineno"> 1472</span>&#160;                                     por(abs_y_is_inf, pandnot(y_is_even, invalid_negative_x))));</div>
<div class="line"><a name="l01473"></a><span class="lineno"> 1473</span>&#160;  <span class="keyword">const</span> Packet pow_is_nan = por(invalid_negative_x, por(x_is_nan, y_is_nan));</div>
<div class="line"><a name="l01474"></a><span class="lineno"> 1474</span>&#160;  <span class="keyword">const</span> Packet pow_is_zero = por(por(por(pand(x_is_zero, y_is_pos),</div>
<div class="line"><a name="l01475"></a><span class="lineno"> 1475</span>&#160;                                         pand(abs_x_is_inf, y_is_neg)),</div>
<div class="line"><a name="l01476"></a><span class="lineno"> 1476</span>&#160;                                     pand(pand(abs_x_is_lt_one, abs_y_is_huge),</div>
<div class="line"><a name="l01477"></a><span class="lineno"> 1477</span>&#160;                                          y_is_pos)),</div>
<div class="line"><a name="l01478"></a><span class="lineno"> 1478</span>&#160;                                 pand(pand(abs_x_is_gt_one, abs_y_is_huge),</div>
<div class="line"><a name="l01479"></a><span class="lineno"> 1479</span>&#160;                                      y_is_neg));</div>
<div class="line"><a name="l01480"></a><span class="lineno"> 1480</span>&#160;  <span class="keyword">const</span> Packet pow_is_inf = por(por(por(pand(x_is_zero, y_is_neg),</div>
<div class="line"><a name="l01481"></a><span class="lineno"> 1481</span>&#160;                                        pand(abs_x_is_inf, y_is_pos)),</div>
<div class="line"><a name="l01482"></a><span class="lineno"> 1482</span>&#160;                                    pand(pand(abs_x_is_lt_one, abs_y_is_huge),</div>
<div class="line"><a name="l01483"></a><span class="lineno"> 1483</span>&#160;                                         y_is_neg)),</div>
<div class="line"><a name="l01484"></a><span class="lineno"> 1484</span>&#160;                                pand(pand(abs_x_is_gt_one, abs_y_is_huge),</div>
<div class="line"><a name="l01485"></a><span class="lineno"> 1485</span>&#160;                                     y_is_pos));</div>
<div class="line"><a name="l01486"></a><span class="lineno"> 1486</span>&#160; </div>
<div class="line"><a name="l01487"></a><span class="lineno"> 1487</span>&#160;  <span class="comment">// General computation of pow(x,y) for positive x or negative x and integer y.</span></div>
<div class="line"><a name="l01488"></a><span class="lineno"> 1488</span>&#160;  <span class="keyword">const</span> Packet negate_pow_abs = pandnot(x_is_neg, y_is_even);</div>
<div class="line"><a name="l01489"></a><span class="lineno"> 1489</span>&#160;  <span class="keyword">const</span> Packet pow_abs = generic_pow_impl(abs_x, y);</div>
<div class="line"><a name="l01490"></a><span class="lineno"> 1490</span>&#160;  <span class="keywordflow">return</span> pselect(y_is_one, x,</div>
<div class="line"><a name="l01491"></a><span class="lineno"> 1491</span>&#160;                 pselect(pow_is_one, cst_one,</div>
<div class="line"><a name="l01492"></a><span class="lineno"> 1492</span>&#160;                         pselect(pow_is_nan, cst_nan,</div>
<div class="line"><a name="l01493"></a><span class="lineno"> 1493</span>&#160;                                 pselect(pow_is_inf, cst_pos_inf,</div>
<div class="line"><a name="l01494"></a><span class="lineno"> 1494</span>&#160;                                         pselect(pow_is_zero, cst_zero,</div>
<div class="line"><a name="l01495"></a><span class="lineno"> 1495</span>&#160;                                                 pselect(negate_pow_abs, pnegate(pow_abs), pow_abs))))));</div>
<div class="line"><a name="l01496"></a><span class="lineno"> 1496</span>&#160;}</div>
<div class="line"><a name="l01497"></a><span class="lineno"> 1497</span>&#160; </div>
<div class="line"><a name="l01498"></a><span class="lineno"> 1498</span>&#160; </div>
<div class="line"><a name="l01499"></a><span class="lineno"> 1499</span>&#160; </div>
<div class="line"><a name="l01500"></a><span class="lineno"> 1500</span>&#160;<span class="comment">/* polevl (modified for Eigen)</span></div>
<div class="line"><a name="l01501"></a><span class="lineno"> 1501</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01502"></a><span class="lineno"> 1502</span>&#160;<span class="comment"> *      Evaluate polynomial</span></div>
<div class="line"><a name="l01503"></a><span class="lineno"> 1503</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01504"></a><span class="lineno"> 1504</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01505"></a><span class="lineno"> 1505</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01506"></a><span class="lineno"> 1506</span>&#160;<span class="comment"> * SYNOPSIS:</span></div>
<div class="line"><a name="l01507"></a><span class="lineno"> 1507</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01508"></a><span class="lineno"> 1508</span>&#160;<span class="comment"> * int N;</span></div>
<div class="line"><a name="l01509"></a><span class="lineno"> 1509</span>&#160;<span class="comment"> * Scalar x, y, coef[N+1];</span></div>
<div class="line"><a name="l01510"></a><span class="lineno"> 1510</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01511"></a><span class="lineno"> 1511</span>&#160;<span class="comment"> * y = polevl&lt;decltype(x), N&gt;( x, coef);</span></div>
<div class="line"><a name="l01512"></a><span class="lineno"> 1512</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01513"></a><span class="lineno"> 1513</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01514"></a><span class="lineno"> 1514</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01515"></a><span class="lineno"> 1515</span>&#160;<span class="comment"> * DESCRIPTION:</span></div>
<div class="line"><a name="l01516"></a><span class="lineno"> 1516</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01517"></a><span class="lineno"> 1517</span>&#160;<span class="comment"> * Evaluates polynomial of degree N:</span></div>
<div class="line"><a name="l01518"></a><span class="lineno"> 1518</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01519"></a><span class="lineno"> 1519</span>&#160;<span class="comment"> *                     2          N</span></div>
<div class="line"><a name="l01520"></a><span class="lineno"> 1520</span>&#160;<span class="comment"> * y  =  C  + C x + C x  +...+ C x</span></div>
<div class="line"><a name="l01521"></a><span class="lineno"> 1521</span>&#160;<span class="comment"> *        0    1     2          N</span></div>
<div class="line"><a name="l01522"></a><span class="lineno"> 1522</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01523"></a><span class="lineno"> 1523</span>&#160;<span class="comment"> * Coefficients are stored in reverse order:</span></div>
<div class="line"><a name="l01524"></a><span class="lineno"> 1524</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01525"></a><span class="lineno"> 1525</span>&#160;<span class="comment"> * coef[0] = C  , ..., coef[N] = C  .</span></div>
<div class="line"><a name="l01526"></a><span class="lineno"> 1526</span>&#160;<span class="comment"> *            N                   0</span></div>
<div class="line"><a name="l01527"></a><span class="lineno"> 1527</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01528"></a><span class="lineno"> 1528</span>&#160;<span class="comment"> *  The function p1evl() assumes that coef[N] = 1.0 and is</span></div>
<div class="line"><a name="l01529"></a><span class="lineno"> 1529</span>&#160;<span class="comment"> * omitted from the array.  Its calling arguments are</span></div>
<div class="line"><a name="l01530"></a><span class="lineno"> 1530</span>&#160;<span class="comment"> * otherwise the same as polevl().</span></div>
<div class="line"><a name="l01531"></a><span class="lineno"> 1531</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01532"></a><span class="lineno"> 1532</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01533"></a><span class="lineno"> 1533</span>&#160;<span class="comment"> * The Eigen implementation is templatized.  For best speed, store</span></div>
<div class="line"><a name="l01534"></a><span class="lineno"> 1534</span>&#160;<span class="comment"> * coef as a const array (constexpr), e.g.</span></div>
<div class="line"><a name="l01535"></a><span class="lineno"> 1535</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01536"></a><span class="lineno"> 1536</span>&#160;<span class="comment"> * const double coef[] = {1.0, 2.0, 3.0, ...};</span></div>
<div class="line"><a name="l01537"></a><span class="lineno"> 1537</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01538"></a><span class="lineno"> 1538</span>&#160;<span class="comment"> */</span></div>
<div class="line"><a name="l01539"></a><span class="lineno"> 1539</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> Packet, <span class="keywordtype">int</span> N&gt;</div>
<div class="line"><a name="l01540"></a><span class="lineno"> 1540</span>&#160;<span class="keyword">struct </span>ppolevl {</div>
<div class="line"><a name="l01541"></a><span class="lineno"> 1541</span>&#160;  <span class="keyword">static</span> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run(<span class="keyword">const</span> Packet&amp; x, <span class="keyword">const</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::type coeff[]) {</div>
<div class="line"><a name="l01542"></a><span class="lineno"> 1542</span>&#160;    EIGEN_STATIC_ASSERT((N &gt; 0), YOU_MADE_A_PROGRAMMING_MISTAKE);</div>
<div class="line"><a name="l01543"></a><span class="lineno"> 1543</span>&#160;    <span class="keywordflow">return</span> pmadd(ppolevl&lt;Packet, N-1&gt;::run(x, coeff), x, pset1&lt;Packet&gt;(coeff[N]));</div>
<div class="line"><a name="l01544"></a><span class="lineno"> 1544</span>&#160;  }</div>
<div class="line"><a name="l01545"></a><span class="lineno"> 1545</span>&#160;};</div>
<div class="line"><a name="l01546"></a><span class="lineno"> 1546</span>&#160; </div>
<div class="line"><a name="l01547"></a><span class="lineno"> 1547</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> Packet&gt;</div>
<div class="line"><a name="l01548"></a><span class="lineno"> 1548</span>&#160;<span class="keyword">struct </span>ppolevl&lt;Packet, 0&gt; {</div>
<div class="line"><a name="l01549"></a><span class="lineno"> 1549</span>&#160;  <span class="keyword">static</span> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run(<span class="keyword">const</span> Packet&amp; x, <span class="keyword">const</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::type coeff[]) {</div>
<div class="line"><a name="l01550"></a><span class="lineno"> 1550</span>&#160;    EIGEN_UNUSED_VARIABLE(x);</div>
<div class="line"><a name="l01551"></a><span class="lineno"> 1551</span>&#160;    <span class="keywordflow">return</span> pset1&lt;Packet&gt;(coeff[0]);</div>
<div class="line"><a name="l01552"></a><span class="lineno"> 1552</span>&#160;  }</div>
<div class="line"><a name="l01553"></a><span class="lineno"> 1553</span>&#160;};</div>
<div class="line"><a name="l01554"></a><span class="lineno"> 1554</span>&#160; </div>
<div class="line"><a name="l01555"></a><span class="lineno"> 1555</span>&#160;<span class="comment">/* chbevl (modified for Eigen)</span></div>
<div class="line"><a name="l01556"></a><span class="lineno"> 1556</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01557"></a><span class="lineno"> 1557</span>&#160;<span class="comment"> *     Evaluate Chebyshev series</span></div>
<div class="line"><a name="l01558"></a><span class="lineno"> 1558</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01559"></a><span class="lineno"> 1559</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01560"></a><span class="lineno"> 1560</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01561"></a><span class="lineno"> 1561</span>&#160;<span class="comment"> * SYNOPSIS:</span></div>
<div class="line"><a name="l01562"></a><span class="lineno"> 1562</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01563"></a><span class="lineno"> 1563</span>&#160;<span class="comment"> * int N;</span></div>
<div class="line"><a name="l01564"></a><span class="lineno"> 1564</span>&#160;<span class="comment"> * Scalar x, y, coef[N], chebevl();</span></div>
<div class="line"><a name="l01565"></a><span class="lineno"> 1565</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01566"></a><span class="lineno"> 1566</span>&#160;<span class="comment"> * y = chbevl( x, coef, N );</span></div>
<div class="line"><a name="l01567"></a><span class="lineno"> 1567</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01568"></a><span class="lineno"> 1568</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01569"></a><span class="lineno"> 1569</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01570"></a><span class="lineno"> 1570</span>&#160;<span class="comment"> * DESCRIPTION:</span></div>
<div class="line"><a name="l01571"></a><span class="lineno"> 1571</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01572"></a><span class="lineno"> 1572</span>&#160;<span class="comment"> * Evaluates the series</span></div>
<div class="line"><a name="l01573"></a><span class="lineno"> 1573</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01574"></a><span class="lineno"> 1574</span>&#160;<span class="comment"> *        N-1</span></div>
<div class="line"><a name="l01575"></a><span class="lineno"> 1575</span>&#160;<span class="comment"> *         - &#39;</span></div>
<div class="line"><a name="l01576"></a><span class="lineno"> 1576</span>&#160;<span class="comment"> *  y  =   &gt;   coef[i] T (x/2)</span></div>
<div class="line"><a name="l01577"></a><span class="lineno"> 1577</span>&#160;<span class="comment"> *         -            i</span></div>
<div class="line"><a name="l01578"></a><span class="lineno"> 1578</span>&#160;<span class="comment"> *        i=0</span></div>
<div class="line"><a name="l01579"></a><span class="lineno"> 1579</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01580"></a><span class="lineno"> 1580</span>&#160;<span class="comment"> * of Chebyshev polynomials Ti at argument x/2.</span></div>
<div class="line"><a name="l01581"></a><span class="lineno"> 1581</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01582"></a><span class="lineno"> 1582</span>&#160;<span class="comment"> * Coefficients are stored in reverse order, i.e. the zero</span></div>
<div class="line"><a name="l01583"></a><span class="lineno"> 1583</span>&#160;<span class="comment"> * order term is last in the array.  Note N is the number of</span></div>
<div class="line"><a name="l01584"></a><span class="lineno"> 1584</span>&#160;<span class="comment"> * coefficients, not the order.</span></div>
<div class="line"><a name="l01585"></a><span class="lineno"> 1585</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01586"></a><span class="lineno"> 1586</span>&#160;<span class="comment"> * If coefficients are for the interval a to b, x must</span></div>
<div class="line"><a name="l01587"></a><span class="lineno"> 1587</span>&#160;<span class="comment"> * have been transformed to x -&gt; 2(2x - b - a)/(b-a) before</span></div>
<div class="line"><a name="l01588"></a><span class="lineno"> 1588</span>&#160;<span class="comment"> * entering the routine.  This maps x from (a, b) to (-1, 1),</span></div>
<div class="line"><a name="l01589"></a><span class="lineno"> 1589</span>&#160;<span class="comment"> * over which the Chebyshev polynomials are defined.</span></div>
<div class="line"><a name="l01590"></a><span class="lineno"> 1590</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01591"></a><span class="lineno"> 1591</span>&#160;<span class="comment"> * If the coefficients are for the inverted interval, in</span></div>
<div class="line"><a name="l01592"></a><span class="lineno"> 1592</span>&#160;<span class="comment"> * which (a, b) is mapped to (1/b, 1/a), the transformation</span></div>
<div class="line"><a name="l01593"></a><span class="lineno"> 1593</span>&#160;<span class="comment"> * required is x -&gt; 2(2ab/x - b - a)/(b-a).  If b is infinity,</span></div>
<div class="line"><a name="l01594"></a><span class="lineno"> 1594</span>&#160;<span class="comment"> * this becomes x -&gt; 4a/x - 1.</span></div>
<div class="line"><a name="l01595"></a><span class="lineno"> 1595</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01596"></a><span class="lineno"> 1596</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01597"></a><span class="lineno"> 1597</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01598"></a><span class="lineno"> 1598</span>&#160;<span class="comment"> * SPEED:</span></div>
<div class="line"><a name="l01599"></a><span class="lineno"> 1599</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01600"></a><span class="lineno"> 1600</span>&#160;<span class="comment"> * Taking advantage of the recurrence properties of the</span></div>
<div class="line"><a name="l01601"></a><span class="lineno"> 1601</span>&#160;<span class="comment"> * Chebyshev polynomials, the routine requires one more</span></div>
<div class="line"><a name="l01602"></a><span class="lineno"> 1602</span>&#160;<span class="comment"> * addition per loop than evaluating a nested polynomial of</span></div>
<div class="line"><a name="l01603"></a><span class="lineno"> 1603</span>&#160;<span class="comment"> * the same degree.</span></div>
<div class="line"><a name="l01604"></a><span class="lineno"> 1604</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l01605"></a><span class="lineno"> 1605</span>&#160;<span class="comment"> */</span></div>
<div class="line"><a name="l01606"></a><span class="lineno"> 1606</span>&#160; </div>
<div class="line"><a name="l01607"></a><span class="lineno"> 1607</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> Packet, <span class="keywordtype">int</span> N&gt;</div>
<div class="line"><a name="l01608"></a><span class="lineno"> 1608</span>&#160;<span class="keyword">struct </span>pchebevl {</div>
<div class="line"><a name="l01609"></a><span class="lineno"> 1609</span>&#160;  EIGEN_DEVICE_FUNC</div>
<div class="line"><a name="l01610"></a><span class="lineno"> 1610</span>&#160;  <span class="keyword">static</span> EIGEN_STRONG_INLINE Packet run(Packet x, <span class="keyword">const</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::type coef[]) {</div>
<div class="line"><a name="l01611"></a><span class="lineno"> 1611</span>&#160;    <span class="keyword">typedef</span> <span class="keyword">typename</span> unpacket_traits&lt;Packet&gt;::type Scalar;</div>
<div class="line"><a name="l01612"></a><span class="lineno"> 1612</span>&#160;    Packet b0 = pset1&lt;Packet&gt;(coef[0]);</div>
<div class="line"><a name="l01613"></a><span class="lineno"> 1613</span>&#160;    Packet b1 = pset1&lt;Packet&gt;(<span class="keyword">static_cast&lt;</span>Scalar<span class="keyword">&gt;</span>(0.f));</div>
<div class="line"><a name="l01614"></a><span class="lineno"> 1614</span>&#160;    Packet b2;</div>
<div class="line"><a name="l01615"></a><span class="lineno"> 1615</span>&#160; </div>
<div class="line"><a name="l01616"></a><span class="lineno"> 1616</span>&#160;    <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 1; i &lt; N; i++) {</div>
<div class="line"><a name="l01617"></a><span class="lineno"> 1617</span>&#160;      b2 = b1;</div>
<div class="line"><a name="l01618"></a><span class="lineno"> 1618</span>&#160;      b1 = b0;</div>
<div class="line"><a name="l01619"></a><span class="lineno"> 1619</span>&#160;      b0 = psub(pmadd(x, b1, pset1&lt;Packet&gt;(coef[i])), b2);</div>
<div class="line"><a name="l01620"></a><span class="lineno"> 1620</span>&#160;    }</div>
<div class="line"><a name="l01621"></a><span class="lineno"> 1621</span>&#160; </div>
<div class="line"><a name="l01622"></a><span class="lineno"> 1622</span>&#160;    <span class="keywordflow">return</span> pmul(pset1&lt;Packet&gt;(<span class="keyword">static_cast&lt;</span>Scalar<span class="keyword">&gt;</span>(0.5f)), psub(b0, b2));</div>
<div class="line"><a name="l01623"></a><span class="lineno"> 1623</span>&#160;  }</div>
<div class="line"><a name="l01624"></a><span class="lineno"> 1624</span>&#160;};</div>
<div class="line"><a name="l01625"></a><span class="lineno"> 1625</span>&#160; </div>
<div class="line"><a name="l01626"></a><span class="lineno"> 1626</span>&#160;} <span class="comment">// end namespace internal</span></div>
<div class="line"><a name="l01627"></a><span class="lineno"> 1627</span>&#160;} <span class="comment">// end namespace Eigen</span></div>
<div class="line"><a name="l01628"></a><span class="lineno"> 1628</span>&#160; </div>
<div class="line"><a name="l01629"></a><span class="lineno"> 1629</span>&#160;<span class="preprocessor">#endif </span><span class="comment">// EIGEN_ARCH_GENERIC_PACKET_MATH_FUNCTIONS_H</span></div>
<div class="ttc" id="anamespaceEigen_html"><div class="ttname"><a href="namespaceEigen.html">Eigen</a></div><div class="ttdoc">Namespace containing all symbols from the Eigen library.</div><div class="ttdef"><b>Definition:</b> Core:139</div></div>
<div class="ttc" id="anamespaceEigen_html_ae7cb2544e4e745bc0067fe793e3f2f81"><div class="ttname"><a href="namespaceEigen.html#ae7cb2544e4e745bc0067fe793e3f2f81">Eigen::expm1</a></div><div class="ttdeci">const Eigen::CwiseUnaryOp&lt; Eigen::internal::scalar_expm1_op&lt; typename Derived::Scalar &gt;, const Derived &gt; expm1(const Eigen::ArrayBase&lt; Derived &gt; &amp;x)</div></div>
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